Motivic Serre group and Sato--Tate conjecture

TL;DR

This paper advances the understanding of the Sato--Tate conjecture by extending the algebraic framework to more general motivic categories and weights, facilitating explicit computation of Sato--Tate groups.

Contribution

It generalizes the algebraic approach to the Sato--Tate conjecture to include broader motivic categories and weights, and refines the connection between $l$-adic representations and motives.

Findings

Extended the algebraic framework to general weights and motivic categories.

Provided new results in the odd weight case.

Facilitated explicit computation of Sato--Tate groups.

Abstract

This paper concerns the Algebraic Sato--Tate and Sato--Tate conjectures, based on Serre's original motivic formulation, with an eye towards explicit computations of Sato--Tate groups. We build on the algebraic framework for the Sato--Tate conjecture introduced in a previous paper, which used Deligne's motivic category for absolute Hodge cycles and was restricted to motives of odd weight. Here, we allow general weight and some other motivic categories, notably Andr\'e's motivic category of motivated cycles; moreover, some results are also new in the odd weight case. The paper consists of two parts; in the first part we work in the framework of strongly compatible families of $l$-adic representations associated with pure, rational, polarized Hodge structures, while in the second part we use the language of motives.

Equations

$$i_{CC}:{\pi }_0(G_{l,K,1}^{\mathrm{alg}})\to {\pi }_0(G_{l,K}^{\mathrm{alg}}).$$

$${\tilde{i}}_{CC}:{\pi }_0(\widetilde{G_{l,K,1}^{\mathrm{alg}}})\stackrel{\simeq }{⟶}{\pi }_0(\widetilde{G_{l,K}^{\mathrm{alg}}}).$$

$$(G_{l,K,1}^{\mathrm{alg}}{)}^{\circ }=\widetilde{G_{l,K_0,1}^{\mathrm{alg}}}.$$

$${\tilde{i}}_{CC}:{\pi }_0(\widetilde{{\mathcal{G}}_{K,1}^{\mathrm{alg}}})\stackrel{\simeq }{⟶}{\pi }_0(\widetilde{{\mathcal{G}}_K^{\mathrm{alg}}}).$$

$$G_{{\mathcal{M}}_{\sim }(M),1}\subset D{L}_K(V,\psi ,D).$$

$$i_M:{\pi }_0(G_{{\mathcal{M}}_{\sim }(M),1})\to {\pi }_0(G_{{\mathcal{M}}_{\sim }(M)}).$$

$${\mathrm{AST}}_K(M):={\mathrm{MS}}_{\sim }(M)$$

$${\mathrm{AST}}_K(M)$$

$$H(V,\psi )$$

$${\pi }_0({\mathrm{AST}}_K(M))$$

$$G_{l,K,1}^{\mathrm{alg}}$$

$${\mathrm{AST}}_K(M{)}^{\circ }$$

$${\pi }_0({\mathrm{AST}}_K(M))$$

$${\mathrm{AST}}_K(M)$$

$$i_{M\oplus \mathbb{T}}:{\pi }_0(G_{{\mathcal{M}}_{\sim }(M\oplus \mathbb{T}),1})\stackrel{\simeq }{⟶}{\pi }_0(G_{{\mathcal{M}}_{\sim }(M\oplus \mathbb{T})}).$$

$$\widetilde{{\mathrm{AST}}_K}(M):={\mathrm{MS}}_{\sim }(M\oplus \mathbb{T}).$$

$${\mathrm{AST}}_K(M{)}^{\circ }$$

$${\mathrm{AST}}_K(M)$$

$${\mathrm{ST}}_K(M)$$

$$\psi :V\times V\to \mathbb{Q}(-n),$$

$${\mathrm{GIso}}_{(V,\psi )}$$

$${\mathrm{Iso}}_{(V,\psi )}$$

$${\mathrm{GIso}}_{(V,\psi )}={\mathbb{G}}_m{\mathrm{Id}}_V\cdot {\mathrm{Iso}}_{(V,\psi )}.$$

$$\psi (\alpha {\mathrm{Id}}_{V}v,\alpha {\mathrm{Id}}_{V}w)=\psi (\alpha v,\alpha w)={\alpha }^2\psi (v,w).$$

$$\alpha {\mathrm{Id}}_V\in {\mathrm{GIso}}_{(V,\psi )}\mathrm{and}\chi (\alpha {\mathrm{Id}}_V)={\alpha }^2.$$

$${\mathrm{GIso}}_{(V,\psi )}$$

$${\mathrm{Iso}}_{(V,\psi )}$$

$$\mathrm{DH}(V,\psi ):=\mathrm{MT}(V,\psi )\cap {\mathrm{Iso}}_{(V,\psi )}.$$

$$H(V,\psi )=\mathrm{DH}(V,\psi {)}^{\circ }.$$

$$1$$

$${\mathbb{G}}_m{\mathrm{Id}}_V\times H(V,\psi )\to \mathrm{MT}(V,\psi ),$$

$$(g_1,g_2)\mapsto g_1{g}_2,$$

$${\mathbb{G}}_m{\mathrm{Id}}_V\cdot H(V,\psi )={\mathbb{G}}_m{\mathrm{Id}}_V\cdot gH(V,\psi )=\mathrm{MT}(V,\psi ).$$

$$\alpha {\mathrm{Id}}_V\cdot g_1=\beta {\mathrm{Id}}_V\cdot g{g}_2.$$

$$1⟶G_0(\mathbb{C})⟶G(\mathbb{C})\stackrel{\pi }{⟶}{\mathbb{G}}_m(\mathbb{C})⟶1.$$

Full text

Thanks to Yves André for answering our questions about motivated cycles, particularly concerning Remark 13.7. Banaszak was supported by UC San Diego (Sept. 2014–June 2015), the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund and Weizmann Institute of Science, Rehovot (Aug. 7–22, 2022). Kedlaya was supported by NSF (grants DMS-1101343, DMS-1501214, DMS-1802161, DMS-2053473), UC San Diego (Stefan E. Warschawski professorship), and the IAS School of Mathematics (Visiting Professorship 2018–2019), and benefitted from the hospitality of IM PAN (Simons Foundation grant 346300, Polish MNiSW fund 2015–2019) during September 2018, MSRI (NSF grants DMS-1440140, DMS-1928930) during May 2019 and January 2023, and CIRM during February 2023.

1. 1 Introduction
2. 2 Mumford–Tate groups of polarized Hodge structures
3. 3 De Rham structures associated with Hodge structures
4. 4 Families of $l$-adic representations associated with Hodge structures
5. 5 The identity connected component of $G_{l,K,1}^{\operatorname{alg}}$
6. 6 Computation of the identity connected component of $G_{l,K,1}^{\operatorname{alg}}$
7. 7 Algebraic Sato–Tate conjecture for families of $l$-adic representations
8. 8 Tate conjecture and Algebraic Sato–Tate conjecture for families of $l$-adic representations
9. 9 The category of polarized realizations $R_{K}^{\rm{p}}$
10. 10 Remarks on equidistribution
11. 11 Application to the Sato–Tate conjecture
12. 12 Relations among motivic categories
13. 13 Assumptions on $\mathcal{M}_{{\sim}}$
14. 14 Motivic Galois group and motivic Serre group
15. 15 Motivic Mumford–Tate group and motivic Serre group
16. 16 The algebraic Sato–Tate group for motives
17. 17 Computation of the identity connected component of $\operatorname{AST}_{K}(M)$
18. 18 Equidistribution of Frobenii in $l$-adic realization of motives

1. Introduction

This paper is a sequel to our previous papers [BK1, BK2]. In those papers, we studied the algebraic underpinning of the generalized Sato–Tate conjecture, which gives a group-theoretic explanation of the distribution of normalized Euler factors of the $L$-function of a motive over a number field. This study provides a framework for rigorous analysis of Sato–Tate groups associated to various classes of motives, which in turn can be used to explain and predict experimental results from the numerical computation of $L$-functions, as in the compilation of the LMFDB (L-Functions and Modular Forms Database) [LMFDB].

The paper [BK1] focused on the case of motives of weight 1, with an eye towards the classification of Sato–Tate groups of abelian surfaces by Fité–Kedlaya–Rotger–Sutherland [FKRS]. The paper [BK2] was devoted to the Sato–Tate conjecture in the framework of Deligne’s motivic category for absolute Hodge cycles, for motives of odd weight. In this paper, we extend the results of [BK2] to other motivic categories and study motives of arbitrary weight; the case of even weight introduces some parity considerations that do not appear for odd weight. Extending to even weight is motivated in part by interest in the Sato–Tate conjecture for K3 surfaces [EJ].

As in [BK2], our approach is based on Serre’s treatment of the generalized Sato–Tate conjecture [Se5] using a construction which we call the motivic Serre group (see [BK2, Def. 9.5]). In [Se5, p. 396], Serre gave the construction in terms of the category $\mathcal{M}_{{\rm{num}}}$ of motives for numerical equivalence (assuming Grothendieck’s standard conjectures and the Hodge conjecture). In [BK2, Def. 11.3], we adapted Serre’s construction for the category $\mathcal{M}_{{\rm{ahc}}}$ of motives for absolute Hodge cycles to define an algebraic Sato–Tate (AST) group associated to such a motive. In this paper, we consider the analogous construction for the category of motives $\mathcal{M}_{{\sim}}$ for a more general equivalence relation ${{\sim}}$, defining an AST group for a homogeneous motive (i.e., a direct summand of a motive of the form $h^{r}_{{\sim}}(X)(m)$); this requires assuming certain conjectures, but for cetain motivic categories these are unconditional (see below).

We now describe the structure of the paper in more detail. The paper is divided into two parts. The first part, §2–11, is concerned with algebraic Sato–Tate groups and the Sato–Tate conjecture for families of $l$-adic representations associated with Hodge structures.

In §2, for a polarized, rational, pure Hodge structure $(V,\psi)$, we recall the definition of the algebraic group $\operatorname{DH}(V,\psi)$ introduced in [BK1, BK2] and its relation to the Hodge group $\operatorname{H}(V,\psi)$, the Mumford–Tate group $\operatorname{MT}(V,\psi)$, and the extended Mumford–Tate group ${\widetilde{\operatorname{MT}}}(V,\psi)$. For a ring $D\subseteq\operatorname{End}_{\mathbb{Q}}(V)$ preserving the Hodge decomposition and having discrete $G_{K}$-module structure (conditions (D1) and (D2)), we also recall the definition of the twisted decomposable Lefschetz group $\operatorname{DL}_{K}(V,\psi,D)$ introduced in [BK1, BK2]. We obtain results concerning the structure and properties of $\operatorname{DH}(V,\psi)$ (Proposition 2.1, Lemma 2.3, Corollary 2.9). We define the Betti parity group $\operatorname{P}(V,\psi)$ and state its basic properties. We give explicit examples for which $\operatorname{P}(V,\psi)$ is nontrivial (Examples 2.15 and 2.16).

In §3 we consider the bilinear space $(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$ over $K$, which has the property that $(V,\psi)\otimes_{\mathbb{Q}}\mathbb{C}\simeq(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})\otimes_{K}\mathbb{C}$. We define the algebraic groups $\operatorname{DH}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$ and $\operatorname{H}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$ assuming that the ring $D$ satisfies additional conditions (DR1) and (DR2). We obtain results concerning the structure and properties of the group $\operatorname{DH}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$ (Proposition 3.1, Lemma 3.2, Corollary 3.5). We define the de Rham parity group $\operatorname{P}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$ and state its basic properties. We give explicit examples, based on Example 2.15, for which $\operatorname{P}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$ is nontrivial (Example 3.10).

In §4, we work with a class of $l$-adic representations associated with Hodge structures from §2 and satisfying additional conditions (R1)–(R4). Examples of such $l$-adic representations are given by étale cohomology of proper and smooth schemes (Remarks 4.1–4.4). We recall the basic properties of the group schemes ${G_{l,K}^{\operatorname{alg}}}$ and ${G_{l,K,1}^{\operatorname{alg}}}$ and the relation of the latter group to the twisted decomposable Lefschetz group (Corollary 4.11). The group scheme ${G_{l,K,1}^{\operatorname{alg}}}$ is fundamental for the setup of the Sato–Tate conjecture.

In §5, we investigate the structure of ${G_{l,K,1}^{\operatorname{alg}}}$ in relation with the structure of ${G_{l,K}^{\operatorname{alg}}}$. The basic results are Theorem 5.1 and Corollary 5.2. There is a natural map between component groups:

[TABLE]

Based upon two technical results (Theorems 5.4 and 5.5) we prove that $i_{CC}$ is an isomorphism when $n$ is odd (Theorem 5.6). For an extension $M_{0}/K$ such that ${G_{l,M_{0}}^{\operatorname{alg}}}$ is connected, we prove two technical results (Propositions 5.7 and 5.8) concerning the structure of ${G_{l,M_{0},1}^{\operatorname{alg}}}$ with relation to its identity component. Let $K_{0}/K$ be the smallest extension such that ${G_{l,K_{0}}^{\operatorname{alg}}}$ is connected; we establish some results concerning ${G_{l,K_{0},1}^{\operatorname{alg}}}$ and the map $i_{CC}$ (Theorem 5.10 and Corollary 5.11). We also define the Serre $l$-adic parity group $P_{\rm{S}}(V_{l},\psi_{l})$ and describe its basic properties. Again we give explicit examples, based on Example 2.15, for which $P_{\rm{S}}(V_{l},\psi_{l})$ is nontrivial (Example 5.16). It follows from Theorem 5.10 that the map $i_{CC}$ is an isomorphism if and only if $P_{\rm{S}}(V_{l},\psi_{l})$ is trivial. At the end of this section we give conditions for $\overline{\rho_{l}(G_{K})_{1}}\,=\,{G_{l,K,1}^{\operatorname{alg}}}$ (Theorem 5.20 and Corollary 5.21).

In §6, following an approach of Serre, we define $\widetilde{G_{l,K}^{\operatorname{alg}}}$ and $\widetilde{G_{l,K,1}^{\operatorname{alg}}}$. We prove that $\widetilde{G_{l,K,1}^{\operatorname{alg}}}\subset G_{l,K,1}^{\operatorname{alg}}$ with index at most 2 and $(\widetilde{G_{l,K,1}^{\operatorname{alg}}})^{\circ}=(G_{l,K,1}^{\operatorname{alg}})^{\circ}$. We also prove that for any weight $n$ the following map is an isomorphism:

[TABLE]

The identity connected component of $G_{l,K,1}^{\operatorname{alg}}$ (Corollary 6.5) is computed as follows:

[TABLE]

We also prove that $G_{l,K,1}^{\operatorname{alg}}=\widetilde{G_{l,K,1}^{\operatorname{alg}}}$ if the parity group $\operatorname{P}(V_{l},\psi_{l})$ is trivial.

In §7, we formulate Algebraic Sato–Tate Conjecture 7.5 and Sato–Tate Conjecture 7.12 for $l$-adic representations with respect to the groups $G_{l,K,1}^{\operatorname{alg}}$. Under Algebraic Sato–Tate Conjecture 7.5, we determine the relationship between the algebraic Sato–Tate and Sato–Tate groups for the fields $K$ and $K_{0}$. We also formulate Algebraic Sato–Tate Conjecture 7.19 and Sato–Tate Conjecture 7.33 with respect to the groups $\widetilde{G_{l,K,1}^{\operatorname{alg}}}$. We prove that Conjecture 7.19 implies Conjecture 7.5 and we discuss obstructions to the opposite implication. We also analyze the relationship between the algebraic Sato–Tate groups $\operatorname{AST}_{K}(V,\psi)$, $\widetilde{\operatorname{AST}_{K}}(V,\psi)$ and the Sato–Tate groups $\operatorname{ST}_{K}(V,\psi)$, $\widetilde{\operatorname{ST}_{K}}(V,\psi)$ (Remark 7.38, Corollary 7.39). Under Conjecture 7.19, we prove (Theorem 7.43) that if $-{{\rm Id}}_{V}\notin\widetilde{\operatorname{ST}_{K}}(V,\psi)$ then Sato–Tate Conjecture 7.12 does not hold, whereas if $-{{\rm Id}}_{V}\in\widetilde{\operatorname{ST}_{K}}(V,\psi)$ then Sato–Tate Conjectures 7.12 and 7.33 are equivalent; this suggests that the latter is the more correct formulation.

In §8 we state Conjectures 8.1 and 8.2 which are $l$-adic analogues of the Tate conjecture for motives. For a special case of these conjectures see [LP]. The difference between the Tate conjecture for motives and these conjectures is the existence of a global object (motivic Galois group) in the Tate conjecture, whereas the existence of global objects $\mathcal{G}_{K}^{\operatorname{alg}}:=\mathcal{G}_{K}^{\operatorname{alg}}(V,\psi)$ and $\widetilde{\mathcal{G}_{K}^{\operatorname{alg}}}:=\widetilde{\mathcal{G}_{K}^{\operatorname{alg}}}(V,\psi)$ is assumed in Conjectures 8.1 and 8.2. Throughout §8 we assume only the weaker form of Conjectures 8.1 and 8.2, namely Conjectures 8.1(a) and 8.2(a); all results in §8 are under this assumption. This framework allows to define naturally Algebraic Sato–Tate groups $\operatorname{AST}_{K}(V,\psi):=\mathcal{G}_{K,1}^{\operatorname{alg}}$ and $\widetilde{\operatorname{AST}_{K}}(V,\psi):=\widetilde{\mathcal{G}_{K,1}^{\operatorname{alg}}}$; the results we obtain are very similar to the results of §6 for the $l$-adic site and to the results of §16 for the motivic site. In particular, among a number of other results, we prove that there is a natural isomorphism (Theorem 8.13):

[TABLE]

Moreover we prove that all four Conjectures 7.5, 7.19, 8.1, 8.2 are equivalent. The framework of this section provides a wider perspective on Algebraic Sato–Tate Conjectures 7.5 and 7.19.

In §9 we present families $(\rho_{l})_{l}$ of $l$-adic representations which are associated with pure, polarized, rational Hodge structures and for which Conjectures 8.1(a) and 8.2(a) are satisfied (see Proposition 9.13). These families will be determined by certain objects in the category of polarized realizations $R_{K}^{\rm{p}}$ which we define; it is a full subcategory of the category of Jannsen realizations $R_{K}$ [Ja1]. For the convenience of readers our computations concerning polarized realizations are very detailed.

§10 is independent of the previous sections and somewhat different in character. In §10 we consider an open subgroup $G_{0}$ of a compact group $G$ and the map $X(j)\colon X(G_{0})\rightarrow X(G)$ of the conjugacy classes induced by natural embedding $j\colon G_{0}\rightarrow G$. Consider a sequence of elements $(x_{n})$ in $X(G)$. Our first result, Proposition 10.7, shows that equidistribution of the sequence $(x_{n})$ in $X(G)$ with respect to the Haar measure $\mu$ on $G$ implies, under a natural numerical condition (10.5), the equidistribution with respect to the pushforward measure $j_{\ast}\mu_{G_{0}}$ of the subsequence $(x_{n_{k}})$ of $(x_{n})$ with elements in the image under the map $X(j)$. For the rest of §10, we study the problem of inverting Proposition 10.7, i.e. to deduce an equidistribution statement in $X(G)$ from equidistribution statements in $X(G_{0})$ and $X(G/G_{0})$. As first results in this direction we prove Lemma 10.11 and Proposition 10.12. To get stronger results concerning deduction of equidistribution on $G$ from equidistribution on $G_{0}$, we first extend Artin’s Theorem on induced characters from finite groups to compact groups (Lemma 10.18). We then prove Lemma 10.19 and Theorem 10.20; in the latter, $G_{0}$ is a connected Lie group satisfying some extra conditions on its factors.

The first part of §11, ending at the proof of Lemma 11.4, depends only on §10. In the first part of §11 we discuss equidistribution on a compact group $G$ via $L$-series associated with representations of $G$ on finite dimensional $\mathbb{C}$-vector spaces, following the idea of Serre [Se1, Appendix to Chapter I]. Consider a sequence $(x_{v})$ of elements of $X(G)$ indexed by prime ideals of $\mathcal{O}_{K}$ (see loc. cit.). Fix any finite extension $L/K$. We prove an auxiliary result (Lemma 11.3) that $(x_{v})$ is equidistributed on $X(G)$ with respect to the measure induced on $X(G)$ by the Haar measure $\mu$ on $G$ if and only if the subsequence $(x_{v})_{S}$ of $(x_{v})$, indexed by $v$’s that split completely in $\mathcal{O}_{L}$, is equidistributed on $X(G)$ with respect to the same measure. We include an auxiliary Lemma 11.4 concerning unipotent elements in algebraic groups which is indispensable in further computations in this section.

In the second part of §11, we assume Conjectures 7.5 and 7.19, that is, we work under the assumptions of Theorems 7.17 and 7.26. We investigate equidistribution of Frobenius elements in the conjugacy classes $X(G)$ of $G:=\operatorname{ST}_{K}(V,\psi)$ and its relationship with equidistribution of Frobenius elements in the conjugacy classes $X(G_{0})$ of $G_{0}:=\operatorname{ST}_{K_{0}}(V,\psi)$. The main result in this section (Theorem 11.12) states that the sequence of Frobenius elements in $X(G)$ is equidistributed if and only if the sequence of Frobenius elements in $X(G_{1})$ is equidistributed for each subgroup $G_{1}$ of $G$ such that $G_{0}\subset G_{1}$ and $G_{1}/G_{0}$ is cyclic. The proof of Theorem 11.12 is based on a number of technical results from previous sections and three technical results from this section: Lemmas 11.8, 11.9, and Corollary 11.10.

Observe that under the assumption of Algebraic Sato–Tate Conjecture 7.5, we have $\operatorname{ST}_{K_{0}}(V,\psi)=\operatorname{ST}_{K}(V,\psi)^{\circ}$ if and only if the Serre $l$-adic parity group $P_{\rm{S}}(V_{l},\psi_{l})$ is trivial (cf. Proposition 7.9 and Theorem 7.14). Hence in the case where $\operatorname{ST}_{K_{0}}(V,\psi)=\operatorname{ST}_{K}(V,\psi)^{\circ}$, Theorem 11.12 states that it is enough to check the Sato–Tate Conjecture 7.12 on cyclic extensions of the identity component of the Sato–Tate group $\operatorname{ST}_{K}(V,\psi)$. Because $\widetilde{\operatorname{ST}_{K_{0}}}(V,\psi)=\widetilde{\operatorname{ST}_{K}}(V,\psi)^{\circ}$ by Theorem 7.23 and Corollary 7.31 then again Theorem 11.12 states that it is enough to check the Sato–Tate Conjecture 7.33 on cyclic extensions of the identity component of the Sato–Tate group $\widetilde{\operatorname{ST}_{K}}(V,\psi)$.

The second part of the paper, §12–18, is concerned with the algebraic Sato–Tate and Sato–Tate conjectures for Hodge structures and $l$-adic representations arising from motives in various motivic categories.

In §12, we give a very brief review of classical motivic categories and relations among them.

In §13, we describe the basic assumptions on the motivic category $\mathcal{M}_{{\sim}}$ that we need to impose in §13–18. In Assumptions 1–3, we require that $\mathcal{M}_{{\sim}}$ satisfies the Chow–Künneth conjecture (leading to motivic Poincare duality), is Tannakian semisimple over $\mathbb{Q}$, and has a motivic star operator. This gives the pure Hodge structure on the Betti realization of a homogeneous motive in $\mathcal{M}_{{\sim}}$. Note that Assumptions 1–3 are known to hold for $\mathcal{M}_{{{\rm{ahc}}}}$ and $\mathcal{M}_{{{\rm{mot}}}}$, the latter being the category of motives for motivated cycles in the sense of André [An2]. We introduce Assumption 4 which gives better control on relations between categories $\mathcal{M}_{K}^{\circ}(D)$ and $\mathcal{M}_{{\sim}}(M)$ (the former one defined in §14); Assumption 4 again holds in $\mathcal{M}_{{\rm{ahc}}}$ and $\mathcal{M}_{{\rm{mot}}}$ (Remark 13.7).

In §14, for a homogeneous motive $M$ in $\mathcal{M}_{\sim}$ and its Betti realization $(V,\psi)$, we define the ring $D:=D(M)$ (see 13.8) with $G_{K}$-discrete module structure and the Artin motive $h^{\circ}(D)$ corresponding to $D$. We also introduce $\mathcal{M}_{K}^{\circ}(D)$, the smallest Tannakian subcategory of the category of Artin motives of $\mathcal{M}_{K}^{\circ}$ containing $h^{\circ}(D)$, and the corresponding motivic Galois group $G_{\mathcal{M}_{K}^{\circ}(D)}$. We recall the motivic Galois groups $G_{\mathcal{M}_{{\sim}}}$ and $G_{\mathcal{M}_{{\sim}}(M)}$. We finally recall the definition of the motivic Serre group $G_{\mathcal{M}_{\sim}(M),1}$ and show the following formula (see 14.13):

[TABLE]

In §15, we investigate the structure of $G_{\mathcal{M}_{\sim}(M),1}$ in relation with the structure of $G_{\mathcal{M}_{{\sim}}(M)}$. There is a natural map between component groups:

[TABLE]

The basic results concerning $i_{M}$ are Theorem 15.2, Corollary 15.3, and Corollary 15.4. In Proposition 15.6, we analyze the relationship between $G_{\mathcal{M}_{{\sim}}(M),1}$ and its identity component. It follows from Theorem 15.2 and Lemma 15.7 that for $n$ odd, the map $i_{M}$ is an isomorphism. We recall Serre’s Conjecture concerning the Motivic Mumford–Tate group and state Motivic Mumford–Tate and Motivic Sato–Tate conjectures. We define the Serre motivic parity group $\operatorname{P}_{\rm{S}}(M,\psi_{{\sim}})$ and prove its basic properties. Once more, we give explicit examples, based on Example 2.15, for which $\operatorname{P}_{\rm{S}}(M,\psi_{{\sim}})$ is nontrivial (Example 15.19). We immediately observe (see Theorem 15.2) that $i_{M}$ is an isomorphism if and only if $\operatorname{P}_{\rm{S}}(M,\psi_{{\sim}})$ is trivial. At the end of §15 we discuss relations among the parity groups we have defined. The Betti, De Rham, and $l$-adic realizations of the motivic category $\mathcal{M}_{{\sim}}$ give natural homomorphisms among these groups. It is immediate from definitions that every parity group is either trivial or isomorphic to $\{\pm 1\}$. We collect from previous sections some general conditions for all parity groups to be trivial (Proposition 15.23(a)) and for all parity groups to be nontrivial (Proposition 15.23(b)).

In §16, following Serre [Se5, sec. 13, pp. 396–397] we define the algebraic Sato–Tate group for a homogeneous motive $M$ as the motivic Serre group (Definition 16.7):

[TABLE]

and the Sato–Tate group $\operatorname{ST}_{K}(M)$ as a maximal compact subgroup of $\operatorname{AST}_{K}(M)(\mathbb{C})$. Theorem 16.8 states that $\operatorname{AST}_{K}(M)$ is reductive and:

[TABLE]

Under the assumption that $\operatorname{H}(V,\psi)=C_{D}(\operatorname{Iso}_{(V,\psi)})$ (i.e., the Hodge group is explained by the endomorphisms in $D$) we determine the structure of $\operatorname{AST}_{K}(M)$ (Corollary 16.11):

[TABLE]

In addition, if the Mumford–Tate conjecture holds for $M$, then the algebraic Sato–Tate conjecture holds for $M$ and the structure of $G_{l,K,1}^{\operatorname{alg}}$ is also determined (Corollary 16.12).

In §17 for a polarized motive $M$ and the Tate motive $\mathbb{T}$ we investigate $G_{\mathcal{M}_{{\sim}}(M\oplus\mathbb{T})}$ and the motivic Serre group $\operatorname{MS}_{{\sim}}(M\oplus\mathbb{T}):=G_{\mathcal{M}_{{\sim}}(M\oplus\mathbb{T}),1}={\rm{Ker}}\,(G_{\mathcal{M}_{{\sim}}(M\oplus\mathbb{T})}\,\,{\stackrel{{\scriptstyle N}}{{\longrightarrow}}}\,\,\mathbb{G}_{m})$. We prove that there is a natural isomorphism (Theorem 17.2):

[TABLE]

We define the algebraic Sato–Tate group:

[TABLE]

Every maximal compact subgroup of $\widetilde{\operatorname{AST}_{K}}(M)(\mathbb{C})$ will be called a Sato–Tate group associated with $M$ and denoted $\widetilde{\operatorname{ST}_{K}}(M)$. Comparing the groups $\operatorname{AST}_{K}(M)$ and $\widetilde{\operatorname{AST}_{K}}(M)$ we obtain:

[TABLE]

up to conjugation in $\operatorname{AST}_{K}(M)(\mathbb{C})$.

Further we prove that the algebraic Sato–Tate conjecture holds for $\operatorname{AST}_{K}(M)$ if and only if it holds for $\widetilde{\operatorname{AST}_{K}}(M)$ (Prop. 17.6). We prove that the Sato–Tate Conjecture for $\operatorname{ST}_{K}(M)$ implies the Sato–Tate Conjecture for $\widetilde{\operatorname{ST}_{K}}(M)$ (Prop. 17.7). In connection with §9 we propose the Geometricity Conjecture 17.12. At the end of §17 we discuss the Sato–Tate parity group $\operatorname{P}_{\rm{ST}}(M,\psi_{{\sim}})$ for motives $M$ and its impact on the Sato–Tate conjecture (Theorem 17.16).

In §18 we consider the Sato–Tate conjecture for homogeneous motives $M$ (direct factors of $h_{{\sim}}^{r}(X)(m)$) of nonzero weight $n=2m-r.$ We assume that the family $(\rho_{l}),$ associated with the $l$-adic realizations of $M,$ is strictly compatible. Then under some additional technical assumptions, the main result in this section (Theorem 18.1) states that the Sato–Tate Conjecture holds for the representation $\rho_{l}\colon G_{K}\rightarrow\operatorname{GIso}_{(V_{l},\psi_{l})}(\mathbb{Q}_{l})$ (resp. $\widetilde{\rho}_{l}\colon G_{K}\rightarrow\operatorname{GL}(V_{l}\oplus\mathbb{Q}_{l}(1))$) with respect to $\operatorname{ST}_{K}(M)$ (resp. $\widetilde{\operatorname{ST}_{K}}(M)$) if and only if it holds for $\rho_{l}\colon G_{K_{1}}\rightarrow\operatorname{GIso}_{(V_{l},\psi_{l})}(\mathbb{Q}_{l})$ (resp. $\widetilde{\rho}_{l}\colon G_{K_{1}}\rightarrow\operatorname{GL}(V_{l}\oplus\mathbb{Q}_{l}(1))$) with respect to $\operatorname{ST}_{K_{1}}(M)$ (resp. $\widetilde{\operatorname{ST}_{K_{1}}}(M)$) for all subextensions $K_{1}$ of $K_{0}/K$ such that $K_{0}/K_{1}$ is cyclic (cf. Theorem 11.12).

At the end of this section we discuss examples concerning Theorems 10.20 and 18.1. The main observations are in Example 18.6 and Remarks 18.7 and 18.8 concerning abelian surfaces and abelian 3-folds.

2. Mumford–Tate groups of polarized Hodge structures

Let $(V,\psi)$ be a rational, polarized, pure Hodge structure of weight $n$ with polarization:

[TABLE]

(cf. [PS, Definition 2.9]). Following [BK2, Chapter 2] define:

[TABLE]

By (2.1) we observe that $\chi$ becomes a character of $\operatorname{GIso}_{(V,\psi)}$. We call $\chi$ the character of the polarization $\psi$. Observe that $\mathbb{G}_{m}{\rm{Id}}_{V}\subseteq\operatorname{GIso}_{(V,\psi)}$ and

[TABLE]

Indeed, for every $\alpha\in\mathbb{G}_{m,\mathbb{Q}}$ we obtain

[TABLE]

Hence

[TABLE]

By the definition of a polarized Hodge structure, $\psi$ is $(-1)^{n}$-symmetric, hence

[TABLE]

Let $\operatorname{MT}(V,\psi)$ (resp. $\operatorname{H}(V,\psi)$) denote the Mumford–Tate (resp. Hodge) group for $(V,\psi)$. Recall the following definition [BK2, Def. 2.6 (2)]:

[TABLE]

By the definition of $\operatorname{Iso}_{(V,\psi)}$ one observes that [BK2, Def. 2.6 (3)]

[TABLE]

By the definition of a rational Hodge structure, $\mathbb{G}_{m}{\rm{Id}}_{V}\subseteq\operatorname{MT}(V,\psi)$. We thus have the following commutative diagram in which the horizontal arrows are closed immersions and the columns are exact.

Proposition 2.1**.**

The algebraic group $\operatorname{DH}(V,\psi)$ has the following properties:

• (a)

$\mathbb{G}_{m}{\rm{Id}}_{V}\cdot\operatorname{H}(V,\psi)\,=\,\mathbb{G}_{m}{\rm{Id}}_{V}\cdot\operatorname{DH}(V,\psi)\,=\,\operatorname{MT}(V,\psi)$.

• (b)

$\operatorname{DH}(V,\psi)=\operatorname{H}(V,\psi)\,\cup\,-{\rm{Id}}_{V}\cdot\operatorname{H}(V,\psi)$.

• (c)

$-{\rm{Id}}_{V}\in\operatorname{H}(V,\psi)$* iff $\operatorname{H}(V,\psi)=\operatorname{DH}(V,\psi)$.*

• (d)

When $n$ is odd then $-{\rm{Id}}_{V}\,\in\operatorname{H}(V,\psi)=\operatorname{DH}(V,\psi)$.

Proof.

(a) Consider a coset $g\operatorname{H}(V,\psi)$ in $\operatorname{DH}(V,\psi)$. Applying [Hu, Section 7.4, Prop. B(b)] to the homomorphism

[TABLE]

we observe that $\mathbb{G}_{m}{\rm{Id}}_{V}\cdot\operatorname{H}(V,\psi)$ and $\mathbb{G}_{m}{\rm{Id}}_{V}\cdot g\operatorname{H}(V,\psi)$ are closed in $\operatorname{MT}(V,\psi)$. They are also of the same dimension as $\operatorname{H}(V,\psi)$ because the left vertical column in Diagram 2.1 is exact and $\operatorname{DH}(V,\psi)^{\circ}=\operatorname{H}(V,\psi)$.

Because $\operatorname{MT}(V,\psi)$ is an irreducible algebraic group and $\mathbb{G}_{m}{\rm{Id}}_{V}\cdot\operatorname{H}(V,\psi)$ is a closed subgroup of the same dimension, we must have

[TABLE]

(b) From (2.4) there are $\alpha,\beta\in\mathbb{G}_{m}$ and $g_{1},g_{2}\in\operatorname{H}(V,\psi)$ such that:

[TABLE]

Applying $\chi$ to (2.5) we obtain $\alpha^{2}=\beta^{2}$. This implies that $\pm{\rm{Id}}_{V}\cdot g_{1}=gg_{2}$. Hence $g\operatorname{H}(V,\psi)=\operatorname{H}(V,\psi)$ or $g\operatorname{H}(V,\psi)=-{\rm{Id}}_{V}\cdot\operatorname{H}(V,\psi)$.

(c) This follows immediately from (b).

(d) This follows immediately from (c) and Lemma 2.3 below. ∎

Lemma 2.2**.**

Let $L$ be a subfield of $\mathbb{C}$. Let $V$ be a finite dimensional $L$-vector space. Let $G\subset GL(V)$ be a connected algebraic group defined over $L$ and $\dim\,G>0.$ For a nontrivial character $\pi\colon G\rightarrow\mathbb{G}_{m}$ over $L$, put $G_{0}:={\rm{Ker}}\,\pi$. Suppose that there exists a cocharacter $s\colon\mathbb{G}_{m}(\mathbb{C})\rightarrow G(\mathbb{C})$ that splits $\pi$ in the following exact sequence:

[TABLE]

Then $G_{0}$ is connected.

Proof.

Observe that $G(\mathbb{C})$ is a connected, complex Lie group. Take any two points $g_{0}$ and $g_{1}$ in $G_{0}(\mathbb{C})$. There is a path $\alpha(t)\in G(\mathbb{C})$ connecting $g_{0}$ and $g_{1}$, i.e., $\alpha(0)=g_{0}$ and $\alpha(1)=g_{1}$. Define a new path

[TABLE]

Notice that $\beta(t)\in G_{0}(\mathbb{C})$ because

[TABLE]

Also observe that $\beta(0)=g_{0}$ and $\beta(1)=g_{1}$. Hence $\beta(t)$ connects $g_{0}$ and $g_{1}$ in $G_{0}(\mathbb{C})$. It follows that $G_{0}(\mathbb{C})$ is connected, and so $G_{0}$ is connected. ∎

Lemma 2.3**.**

For $n$ odd we have

[TABLE]

Proof.

By the definition of the Mumford–Tate group, we have a cocharacter

[TABLE]

such that for all $z\in\mathbb{G}_{m}(\mathbb{C})$, $\mu_{\infty,V}(z)$ acts by multiplication by $z^{-p}$ on the subspace $V^{p,n-p}$. It follows from the definition of a polarization of a Hodge structure that $\chi\circ\mu_{\infty,V}(z)=z^{-n}$ for every $z\in\mathbb{G}_{m}(\mathbb{C})$. Because $\mathbb{G}_{m}{\rm{Id}}_{V}\subset\operatorname{MT}(V,\psi)$ we have the diagonal cocharacter:

[TABLE]

Because $\chi(w(z))=z^{2}$ for every $z\in\mathbb{G}_{m}(\mathbb{C})$ ([BK2, cf. Remark 2.4]) and $n$ is odd the cocharacter

[TABLE]

is a splitting of $\chi$ in the following exact sequence:

[TABLE]

Observe that $\operatorname{MT}(V,\psi)(\mathbb{C})$ is a connected Lie group because $\operatorname{MT}(V,\psi)$ is a connected algebraic group. By Lemma 2.2, $\operatorname{DH}(V,\psi)$ is connected. Hence the claim of this lemma follows from Proposition 2.1(b). ∎

Consider the cocharacter:

[TABLE]

Definition 2.4**.**

The extended Mumford–Tate group ${\widetilde{\operatorname{MT}}}(V,\psi)$ (cf. [PS, Definition 2.13]) is the smallest subgroup over $\mathbb{Q}$ of the group scheme $\operatorname{GIso}_{(V,\psi)}\times\mathbb{G}_{m}$ such that

[TABLE]

There is a natural embedding

[TABLE]

Applying projections on the first and second factor of $\operatorname{MT}(V,\psi)\times{\mathbb{G}_{m}}$, we obtain natural morphisms $\pi\colon{\widetilde{\operatorname{MT}}}(V,\psi)\rightarrow\operatorname{MT}(V,\psi)$ and $N\colon{\widetilde{\operatorname{MT}}}(V,\psi)\rightarrow\mathbb{G}_{m}$.

Lemma 2.5**.**

There is the following commutative diagram with exact rows and columns. The left vertical arrow is the natural one.

Proof.

Considering Diagram 2.2 on $\mathbb{C}$-points, the morphisms $(x\mapsto x^{-n})\circ N$ and $\chi\circ\pi$ are equal when restricted to the subgroup ${\widetilde{\mu}_{\infty,V}}(\mathbb{G}_{m}(\mathbb{C}))$ of ${\widetilde{\operatorname{MT}}}(V,\psi)(\mathbb{C})$ because of [BK2, (2.9)] and the equality $\pi\circ{\widetilde{\mu}_{\infty,V}}=\mu_{\infty,V}$. The morphisms in the right square of Diagram 2.2 are defined over $\mathbb{Q}$. Hence

[TABLE]

is a closed subgroup of ${\widetilde{\operatorname{MT}}}(V,\psi)$ over $\mathbb{Q}$ whose $\mathbb{C}$-points contain ${\widetilde{\mu}_{\infty,V}}(\mathbb{G}_{m}(\mathbb{C}))$. Hence $\{g\in{\widetilde{\operatorname{MT}}}(V,\psi)\colon(x\mapsto x^{-n})\circ N(g)=\chi\circ\pi(g)\}={\widetilde{\operatorname{MT}}}(V,\psi)$ by definition of the group ${\widetilde{\operatorname{MT}}}(V,\psi)$. Hence the right square of Diagram 2.2 commutes. The map $\pi_{1}$ in this diagram is defined by $\pi$ and clearly makes the left square also commute.

The image $\pi({\widetilde{\operatorname{MT}}}(V,\psi))$ is a closed algebraic subgroup of $\operatorname{MT}(V,\psi)$ over $\mathbb{Q}$ [Hu, Prop. B, section 7.4 cf. section 34.2] containing $\mu_{\infty,V}(\mathbb{G}_{m}(\mathbb{C}))$. Hence, by the definition of $\operatorname{MT}(V,\psi)$, the morphism $\pi$ in Diagram 2.2 is an epimorphism.

Observe that the cocharacter ${\widetilde{\mu}_{\infty,V}}$ splits the morphism $N$ on $\mathbb{C}$-points. Hence by Lemma 2.2 the kernel of $N$ in Diagram 2.2 is connected.

By the definition of $N$ and (2.7), the kernel of $\pi$ in Diagram 2.2 is contained in ${\rm{Id}}_{V}\times\mathbb{G}_{m}$. Hence the restriction of $N$ to the kernel of $\pi$ is a monomorphism. By the commutativity of Diagram 2.2, the kernel of $\pi$ injects into $\mu_{n}$. Hence $\pi_{1}$ is a monomorphism and $\dim{\widetilde{\operatorname{MT}}}(V,\psi)=\dim\operatorname{MT}(V,\psi)$. Consequently $\dim{\rm{Ker}}\,N=\dim\operatorname{DH}(V,\psi)$. This shows that

[TABLE]

as desired. ∎

Let $K$ be a number field. Fix an algebraic closure $\overline{K}$ of $K$. Let $G_{K}:=G(\overline{K}/K)$ denote the absolute Galois group of $K$. We assume from now through §8 that $(V,\psi)$ satisfies the following additional conditions:

• (D1)

There is a ring $D\subseteq\operatorname{End}_{\mathbb{Q}}(V)$ such that the action of $D$ on $V\otimes\mathbb{C}$ preserves the Hodge decomposition, i.e. $DV^{p,q}\subseteq V^{p,q}$ for all $p,q$.

• (D2)

The ring $D$ has a structure of discrete $G_{K}$-module.

Applying (2.3) we obtain

[TABLE]

By (D1) $D$ commutes with $\mu_{\infty,V}(\mathbb{C}^{\times})$ on $V_{\mathbb{C}}$ elementwise, hence the properties of the polarization (cf. [BK2, p. 4]) give:

[TABLE]

By (D2) we obtain the following representation

[TABLE]

Define $K_{e}\,:=\,\overline{K}^{{\rm{Ker}}\rho_{e}}$ (cf. [BK2, Def. 3.1]). For any $\tau\in\operatorname{Gal}(K_{e}/K)$, put

[TABLE]

Observe that

[TABLE]

Definition 2.6**.**

For any $\tau\in\operatorname{Gal}(K_{e}/K)$ put (see [BK2, Def. 3.3]):

[TABLE]

Remark 2.7*.*

The group $\operatorname{DL}_{K}^{\tau}(V,\psi,D)$ is a closed subscheme of $\operatorname{Iso}_{(V,\,\psi)}$ for each $\tau\in\operatorname{Gal}(K_{e}/K)$ and is defined over $\mathbb{Q}$ because $D\subseteq\operatorname{End}_{\mathbb{Q}}(V)$.

Definition 2.8**.**

(Twisted decomposable algebraic Lefschetz group [BK2, Def. 3.4])

For the triple $(V,\,\psi,D)$ define the closed algebraic subgroup of $\operatorname{Iso}_{(V,\,\psi)}$ given by

[TABLE]

From Definition 2.8, there is the following natural monomorphism:

[TABLE]

By (2.9), (2.11), (2.12), and the definition of $\operatorname{MT}(V,\psi)$ we obtain (cf. [BK2, (2.16), (2.17), (2.18)]):

[TABLE]

Corollary 2.9**.**

Assume that $\operatorname{H}(V,\psi)=C_{D}(\operatorname{Iso}_{(V,\psi)})$. Then

[TABLE]

Proof.

This follows from (2.16) and the definition of $\operatorname{H}(V,\psi)$. ∎

Remark 2.10*.*

If $A$ is an abelian variety over $\overline{\mathbb{Q}}$ of dimension 4 considered by Mumford [Mu] and $V=H^{1}(A(\mathbb{C}),\mathbb{Q})$, then $-{\rm{Id}}_{V}\in\operatorname{H}(V,\psi)$ but the condition $\operatorname{H}(V,\psi)=C_{D}(\operatorname{Iso}_{(V,\psi)})$ fails in this case.

Remark 2.11*.*

See [BK2, Chapter 3] for further properties of the twisted decomposable algebraic Lefschetz group.

Definition 2.12**.**

The Betti parity group:

[TABLE]

is the component group of $\operatorname{DH}(V,\psi)$.

Proposition 2.13**.**

If $n$ is odd or if $\operatorname{H}(V,\psi)=C_{D}(\operatorname{Iso}_{(V,\psi)})$ then $\operatorname{P}(V,\psi)$ is trivial.

Proof.

This follows from Lemma 2.3 and Corollary 2.9. ∎

Proposition 2.14**.**

If $n$ is even and ${\rm dim}_{\mathbb{Q}}\,V$ is odd then $\operatorname{P}(V,\psi)$ is nontrivial.

Proof.

Because $n$ is even, $\operatorname{Iso}_{(V,\psi)}=\operatorname{O}_{(V,\psi)}$. Hence $\operatorname{H}(V,\psi)\subseteq\operatorname{Iso}_{(V,\psi)}\subseteq{\pm{\rm{Id}}_{V}}\operatorname{SL}_{V}$. It follows that $-{\rm{Id}}_{V}\notin\operatorname{H}(V,\psi)$ because ${\rm{det}}(-{\rm{Id}}_{V})=(-1)^{{\rm{dim}}_{\mathbb{Q}}\,V}=-1$ and $\operatorname{H}(V,\psi)$ is connected. ∎

Example 2.15**.**

Let $A$ be an abelian threefold over $K$. Let $V:=H^{2}(A(\mathbb{C}),\,\mathbb{Q})=\bigwedge^{2}\,H^{1}(A(\mathbb{C}),\,\mathbb{Q})$. Then $V$ has weight 2 and ${\rm dim}_{\mathbb{Q}}\,V={\binom{6}{2}}=15$. Hence $\operatorname{P}(V,\psi)$ is nontrivial.

Example 2.16**.**

Let $A$ be an elliptic curve over $K$. Let $V:=\operatorname{Sym}^{2}\,H^{1}(A(\mathbb{C}),\,\mathbb{Q})$. Then $V$ has weight 2 and ${\rm dim}_{\mathbb{Q}}\,V=3$. Hence again $\operatorname{P}(V,\psi)$ is nontrivial.

3. De Rham structures associated with Hodge structures

In the following discussion, let $\sigma\colon K\hookrightarrow\mathbb{C}$ denote an arbitrary field embedding. Suppose that we are given a $K$-vector space $V_{{}_{\rm{DR}}}$, a bilinear, nondegenerate pairing

[TABLE]

and a decreasing filtration $F^{i}V_{{}_{\rm{DR}}}$. Suppose in addition that for each $\sigma$ there are isomorphisms:

[TABLE]

[TABLE]

and the vector space $V_{{}_{\rm{DR}}}\otimes_{\sigma}\mathbb{C}$ has a decreasing filtration, induced from $V_{{}_{\rm{DR}}}$, compatible with the Hodge filtration on $V\otimes_{\mathbb{Q}}\mathbb{C}$ via (3.2).

Setting $G^{i}V_{{}_{\rm{DR}}}:=F^{i}V_{{}_{\rm{DR}}}/F^{i+1}V_{{}_{\rm{DR}}}$, we obtain an isomorphism:

[TABLE]

and a Hodge decomposition:

[TABLE]

For the space $(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$ we define:

[TABLE]

We observe that $\chi_{{}_{\rm{DR}}}$ becomes a character of $\operatorname{GIso}_{(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})}$. Moreover $\mathbb{G}_{m,K}{\rm{Id}}_{V_{{}_{\rm{DR}}}}\subseteq\operatorname{GIso}_{(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})}$ and

[TABLE]

For every $\alpha\in\mathbb{G}_{m,K}$ we obtain

[TABLE]

Hence

[TABLE]

Define $\operatorname{MT}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$ and $\operatorname{H}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$ by analogy with the corresponding Mumford–Tate and Hodge groups $\operatorname{MT}(V,\psi)$ and $\operatorname{H}(V,\psi)$. Namely, consider the cocharacter

[TABLE]

such that for any $z\in{\mathbb{C}}^{\times}$, the automorphism $\mu_{{}_{\rm{DR}},V}(z)$ acts as multiplication by ${z}^{-p}$ on $V^{p,q}$ for each $p+q=n$ (cf. (3.4)). Then we define $\operatorname{MT}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$ to be the smallest algebraic subgroup of $\operatorname{GL}_{V_{{}_{\rm{DR}}}}$ over $K$ containing $\mu_{{}_{\rm{DR}},V}(\mathbb{C})$. Define:

[TABLE]

Proposition 3.1**.**

The algebraic group $\operatorname{DH}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$ has the following properties:

• (a)

$\mathbb{G}_{m}{\rm{Id}}_{V_{{}_{\rm{DR}}}}\cdot\operatorname{H}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})\,=\,\mathbb{G}_{m}{\rm{Id}}_{V_{{}_{\rm{DR}}}}\cdot\operatorname{DH}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})\,=\,\operatorname{MT}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$.

• (b)

$\operatorname{DH}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})=\operatorname{H}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})\,\cup\,-{\rm{Id}}_{V_{{}_{\rm{DR}}}}\cdot\operatorname{H}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$.

• (c)

$-{\rm{Id}}_{V_{{}_{\rm{DR}}}}\in\operatorname{H}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$* iff $\operatorname{H}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})=\operatorname{DH}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$.*

• (d)

When $n$ is odd then $-{\rm{Id}}_{V_{{}_{\rm{DR}}}}\,\in\operatorname{H}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})=\operatorname{DH}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$.

Proof.

Similar to the proof of Proposition 2.1. ∎

Lemma 3.2**.**

For $n$ odd,

[TABLE]

Proof.

Similar to the proof of Lemma 2.3. ∎

Lemma 3.3**.**

We have the following inclusions:

[TABLE]

Proof.

By the isomorphisms (3.2), (3.3) and the definition of $\operatorname{MT}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$, we obtain (3.8). Consequently we also get (3.10). Now comparing Proposition 2.1(b) with Proposition 3.1(b) we obtain (3.9). ∎

We assume from now through §8 that $(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$ satisfies the following additional conditions:

• (DR1)

There is a canonical embedding $D\subseteq{{\rm End}}_{\overline{K}}(\overline{V}_{{}_{\rm{DR}}})$ compatible with isomorphisms (3.2) and (3.3), where $\overline{V}_{{}_{\rm{DR}}}:=V_{{}_{\rm{DR}}}\otimes_{K}\overline{K}$ with $\overline{K}$ being the integral closure of $K$ via the chosen embedding in $\mathbb{C}$.

• (DR2)

The action of $D$ preserves the Hodge decomposition (3.5).

By (DR1) and (DR2), $D$ commutes with $\mu_{\infty,V}(\mathbb{C}^{\times})$ on $V_{{}_{\rm{DR}}}\otimes_{K,\sigma}\mathbb{C}$ elementwise. By (3.2)–(3.5) and [BK2, (2.9)] we obtain

[TABLE]

In the De Rham structures we also consider the following subring $D_{e}\subset D$:

[TABLE]

Remark 3.4*.*

Observe that $D_{e}=D^{G(K_{e}/K)}$. Hence if $K=K_{e}$ then $D_{e}=D$.

Obviously $D_{e}$ is compatible with the isomorphisms (3.2) and (3.3), and from (3.11) we obtain:

[TABLE]

Hence

[TABLE]

[TABLE]

Corollary 3.5**.**

Assume that $\operatorname{H}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})=C_{D_{e}}(\operatorname{Iso}_{(V_{{}_{\rm{DR}}},\psi)})$. Then

[TABLE]

Proof.

This follows from (3.15) and the definition of $\operatorname{H}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$. ∎

Definition 3.6**.**

The De Rham parity group:

[TABLE]

is the component group of $\operatorname{DH}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$.

Proposition 3.7**.**

If $n$ is odd or $\operatorname{H}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})=C_{D_{e}}(\operatorname{Iso}_{(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})})$, then $\operatorname{P}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$ is trivial.

Proof.

This follows from Lemma 3.2 and Corollary 3.5. ∎

Corollary 3.8**.**

There is a natural epimorphism:

[TABLE]

Proof.

It follows from Lemma 3.3. ∎

Corollary 3.9**.**

If $n$ is even and ${\rm dim}_{\mathbb{Q}}\,V$ is odd then $\operatorname{P}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$ is nontrivial.

Proof.

It follows from Proposition 2.14 and Corollary 3.8. ∎

Example 3.10**.**

Let $A$ be an abelian threefold over $K$. Put $V_{{}_{\rm{DR}}}:=H^{2}_{{}_{\rm{DR}}}(A)$. Then by Example 2.15 and Corollary 3.9, the group $\operatorname{P}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$ is nontrivial in this case.

Example 3.11**.**

Let $A$ be an elliptic curve over $K$. Put $V_{{}_{\rm{DR}}}:=\operatorname{Sym}^{2}\,H^{1}_{{}_{\rm{DR}}}(A)$. Then by Example 2.16 and Corollary 3.9, the group $\operatorname{P}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$ is again nontrivial.

4. Families of $l$-adic representations associated with Hodge structures

For $(V,\,\psi)$ as in §2 and $l$ a prime number, put $V_{l}:=V\otimes_{\mathbb{Q}}{\mathbb{Q}}_{l}$,   $\psi_{l}:=\psi\otimes_{\mathbb{Q}}\mathbb{Q}_{l}$, and $(V_{l},\psi_{l}):=(V\otimes_{\mathbb{Q}}\mathbb{Q}_{l},\psi\otimes_{\mathbb{Q}}\mathbb{Q}_{l})$. Assume that the bilinear form $\psi_{l}\colon V_{l}\times V_{l}\rightarrow\mathbb{Q}_{l}(-n)$ is $G_{K}$-equivariant. Let

[TABLE]

be the corresponding $l$-adic representation. Observe that:

[TABLE]

Let $\chi_{c}\colon G_{K}\rightarrow\mathbb{Z}_{l}^{\times}$ be the cyclotomic character. Then by the $G_{K}$-equivariance of $\psi_{l}$, we obtain:

[TABLE]

We assume from now through §8 that the family of $l$-adic representations (4.1) satisfy the following additional, natural conditions:

• (R1)

The family $(\rho_{l})_{l}$ is strictly compatible in the sense of Serre [Se1, Chapter I, §2.3], and $\rho_{l}$ is of Hodge–Tate type for every prime of $\mathcal{O}_{K}$ over $l$.

• (R2)

For each $\rho_{l}$, for each prime $v$ of $\mathcal{O}_{K}$ outside of a finite set of primes containing all primes over $l$, the complex absolute values of the eigenvalues of a Frobenius at $v$ are $q_{v}^{-\frac{n}{2}}$.

• (R3)

Let $\lambda$ be any prime in $\mathcal{O}_{K}$ over $l$ and put $\mathbb{C}_{l}:={\widehat{\overline{K}}}_{\lambda}$. For the $K$-vector space $V_{{}_{\rm{DR}}}$ with corresponding structure and the gradation $G^{i}V_{{}_{\rm{DR}}}$ as in §2, there is a corresponding Hodge–Tate decomposition:

[TABLE]

• (R4)

The induced action of $D$ on $V_{l}$ is $G_{K}$-equivariant, i.e., $\forall\beta\in D$, $\forall v_{l}\in V_{l}$, and $\forall\sigma\in G_{K}$,

[TABLE]

Moreover, the induced action of $D$ on $V_{l}\otimes_{{\mathbb{Q}}_{l}}\mathbb{C}_{l}$ is compatible with (4.3).

Remark 4.1*.*

Strictly compatible families of $l$-adic representations of Hodge–Tate type arise naturally from étale cohomology. Indeed, if $X/K$ is a proper scheme and ${\overline{X}}:=X\otimes_{K}{\overline{F}}$, then $V_{l,\operatorname{et}}^{n}:=H^{n}_{\operatorname{et}}({\overline{X}},\,\mathbb{Q}_{l})$ is potentially semistable as a $G_{K_{\lambda}}$-representation for every $\lambda|l$ (see [Ts1, Cor. 2.2.3], [Ts2]). Hence the representation

[TABLE]

is of Hodge–Tate type (cf. [Su, p. 603]). So condition (R1) holds for the family $(\rho_{l,\operatorname{et}}^{n})$.

Remark 4.2*.*

In [BK2, Page 18], when assuming that the complex absolute values of the eigenvalues of a Frobenius element at $v$ are $q_{v}^{\frac{n}{2}}$, we meant the geometric Frobenius. In (R2) above we mean the arithmetic Frobenius.

Remark 4.3*.*

The assumption (R3) holds for families $(\rho_{l,\operatorname{et}}^{n})$ from Remark 4.1 by the following considerations. Let $X/K$ be a smooth projective variety. For a field extension $L/K$, let $X_{L}:=X\otimes_{K}L$. The $\mathbb{Q}$-vector space $V:=H^{n}(X(\mathbb{C}),\,\mathbb{Q})$ admits a pure, polarized Hodge structure of weight $n$:

[TABLE]

where $V^{i,j}:=H^{i}_{{\rm{an}}}(X({\mathbb{C}}),\Omega_{X({\mathbb{C}})}^{j})\simeq H^{i}_{{\rm{zar}}}(X_{\mathbb{C}},\Omega_{X_{\mathbb{C}}}^{j})$. There is the following spectral sequence (cf. [D1, p.17]):

[TABLE]

By [D2, Theorem 5.5] and [Ha, III, Prop. 9.3] we have the following natural isomorphisms:

[TABLE]

By [Fa] we have the following decomposition:

[TABLE]

Observe that $H^{n}_{et}(\overline{{X_{K}}},\,\,\mathbb{Q}_{l})\simeq H^{n}_{et}(\overline{{X_{K_{\lambda}}}},\,\,\mathbb{Q}_{l})$ ([Mi1, Corollary 2.6, Chap. VI]) and

[TABLE]

Remark 4.4*.*

(R4) yields a natural, $G_{K}$-equivariant embedding $D\otimes_{\mathbb{Q}}\,\mathbb{Q}_{l}\,\,\subseteq\operatorname{End}_{\mathbb{Q}_{l}}(V_{l})$.

Recall the following notation from [BK2, pp. 18–19].

Definition 4.5**.**

Let $L/K$ be a finite extension. Let

[TABLE]

be the Zariski closure of $\rho_{l}(G_{L})$ in $\operatorname{GIso}_{(V_{l},\psi_{l})}$. Put:

[TABLE]

For a representation $\rho_{l}$ of Hodge–Tate type, the theorem of Bogomolov on homotheties ([Bo, Théorème 1]; cf. [Su, Prop. 2.8]) applies, meaning that $\rho_{l}(G_{K})\,\cap\,\mathbb{Q}_{l}^{\times}\,{\rm{Id}}_{V_{l}}$ is open in $\mathbb{Q}_{l}^{\times}\,{\rm{Id}}_{V_{l}}$. Therefore $G_{l,L}^{\operatorname{alg}}$ contains the homotheties of $\operatorname{GL}(V_{l})$, and there is an exact sequence

[TABLE]

Remark 4.6*.*

By the theorem of Bogomolov, $\rho_{l}(G_{K})$ is open in $G_{l,K}^{\operatorname{alg}}(\mathbb{Q}_{l})$.

Remark 4.7*.*

By (4.4) $\forall\beta\in D$, $\forall v_{l}\in V_{l}$, and $\forall\sigma\in G_{K}$:

[TABLE]

hence

[TABLE]

Definition 4.8**.**

Put:

[TABLE]

Observe that by (2.12), (4.10), and (4.12) we obtain

[TABLE]

Lemma 4.9**.**

There are natural equalities of group schemes:

[TABLE]

Proof.

Let $\tilde{\tau}\in G_{K}$ be a lift of $\tau\in\operatorname{Gal}(K_{e}/K)$. The coset $\tilde{\tau}\,G_{K_{e}}$ does not depend on the lift. The Zariski closure of $\rho_{l}(\tilde{\tau}\,G_{K_{e}})=\rho_{l}(\tilde{\tau})\,\rho_{l}(G_{K_{e}})$ in $\operatorname{GIso}_{(V_{l},\psi_{l})}$ is $\rho_{l}(\tilde{\tau})\,G_{l,K_{e}}^{\operatorname{alg}}$. Since $\rho_{l}(\tilde{\tau})\,\rho_{l}(G_{K_{e}})\subseteq(G_{l,K}^{\operatorname{alg}})^{\tau}$ then $\rho_{l}(\tilde{\tau})\,G_{l,K_{e}}^{\operatorname{alg}}\subseteq(G_{l,K}^{\operatorname{alg}})^{\tau}$. Because

[TABLE]

we then have

[TABLE]

This implies the equalities (4.17) and (4.18). ∎

Remark 4.10*.*

In the proof of (4.18) we observe that $\rho_{l}(\tilde{\tau})\,G_{l,K_{e}}^{\operatorname{alg}}=(G_{l,K}^{\operatorname{alg}})^{\tau}$ for all $\tau$. Hence we obtain $(G_{l,K,1}^{\operatorname{alg}})^{\operatorname{id}}=G_{l,K_{e},1}^{\operatorname{alg}}$ and a natural isomorphism:

[TABLE]

Corollary 4.11**.**

We have

[TABLE]

Proof.

This follows from (2.13), (4.16) and (4.18). ∎

Because $\operatorname{DL}_{K}^{\operatorname{id}}(V,\psi,D)=\operatorname{DL}_{K_{e}}(V,\psi,D)$ and $(G_{l,K,1}^{\operatorname{alg}})^{\operatorname{id}}=G_{l,K_{e},1}^{\operatorname{alg}}$ we obtain

[TABLE]

Remark 4.12*.*

We are interested in triples $(V,\psi,D)$ for which equality in (4.22) holds for each $l$. In such cases, Conjecture 7.5 below also holds, i.e., $\operatorname{AST}_{K}(V,\psi)=\operatorname{DL}_{K}(V,\psi,D)$. In Corollary 5.3, we prove that equality in (4.22) is equivalent to equality in (4.23).

5. The identity connected component of $G_{l,K,1}^{\operatorname{alg}}$

In this section we extend our results from [BK2, Chapters 2–8] for odd weight to any weight, for families of strictly compatible $l$-adic representations of Hodge–Tate type associated with pure Hodge structures. In [BK2, Section 4] we proved the following theorem:

Theorem 5.1**.**

Let $L/K$ be finite Galois. The following natural map is an isomorphism of finite groups:

[TABLE]

In particular there are the following isomorphisms:

[TABLE]

Proof.

See [BK2, Theorem 4.6]. ∎

Corollary 5.2**.**

There are natural isomorphisms:

[TABLE]

Proof.

This follows by (4.16), (4.21), and Theorem 5.1. ∎

Corollary 5.3**.**

The equality in (4.22) holds if and only if it holds in (4.23).

Proof.

This follows from (5.3) and the equality $(G_{l,K,1}^{\operatorname{alg}})^{\operatorname{id}}=G_{l,K_{e},1}^{\operatorname{alg}}$. ∎

Theorem 5.4**.**

Let $L_{0}/K$ be a finite Galois extension such that $G_{l,L_{0}}^{\operatorname{alg}}$ is connected. There is the following exact sequence:

[TABLE]

Moreover $i_{CC}$ is an isomorphism if and only if $G_{l,L_{0},1}^{\operatorname{alg}}$ is connected.

Proof.

The exactness of (5.4) follows from Theorem 5.1 and the equalities $G_{l,L_{0}}^{\operatorname{alg}}=(G_{l,K}^{\operatorname{alg}})^{\circ}$ and $G_{l,L_{0},1}^{\operatorname{alg}}=G_{l,K,1}^{\operatorname{alg}}\cap G_{l,L_{0}}^{\operatorname{alg}}$. Recall that the group of connected components of an algebraic group is finite. Hence it follows from (5.4) that $G_{l,L_{0},1}^{\operatorname{alg}}$ is connected iff $G_{l,L_{0},1}^{\operatorname{alg}}=(G_{l,K,1}^{\operatorname{alg}})^{\circ}$, but this is equivalent to the statement that $i_{CC}$ is an isomorphism. ∎

Theorem 5.5**.**

Assume that the weight $n$ of $(V,\psi)$ is odd. Then there is a finite Galois extension $L_{0}/K$ such that $G_{l,L_{0}}^{\operatorname{alg}}=(G_{l,K}^{\operatorname{alg}})^{\circ}$ and $G_{l,L_{0},1}^{\operatorname{alg}}=({G_{l,K,1}^{\operatorname{alg}}})^{\circ}$.

Proof.

For the proof of the first equality, see the proof of [BK2, Prop. 4.7]. We refine the proof of the second equality.

Let $L_{0}$ be such that the first equality holds. Restrict the $l$-adic representation to the base field $L_{0}$. Using the Hodge–Tate property of $V_{l}$, after taking $\mathbb{C}_{l}$-points in the exact sequence (4.13) one can apply the homomorphism [Se6, p. 114] defined by Serre:

[TABLE]

such that for all $x\in\mathbb{G}_{m}(\mathbb{C}_{l})$, $h_{l}(x)$ acts by multiplication by $x^{-p}$ on the subspace:

[TABLE]

Choose embeddings of $L_{0}$ into $\mathbb{C}_{l}$ and $\mathbb{C}$ and a compatible isomorphism $\mathbb{C}_{l}\simeq\mathbb{C}$. Extending coefficients to $\mathbb{C}$ using this isomorphism and applying the property (R3), we obtain from $h_{l}$ a homomorphism

[TABLE]

which is defined in the same way as $\mu_{\infty,V}$ of Lemma 2.3 via natural isomorphisms:

[TABLE]

Hence $\chi(h(z))=z^{-n}$ for every $z\in\mathbb{G}_{m}(\mathbb{C})$. Let

[TABLE]

be the diagonal homomorphism; this is well-defined thanks to the comments before Remark 4.6. We know (see §2) that $\chi(w(z))=z^{2}$ for every $z\in\mathbb{G}_{m}(\mathbb{C})$. Hence the homomorphism

[TABLE]

is a splitting of $\chi$ in the following exact sequence:

[TABLE]

Because $G_{l,L_{0}}^{\operatorname{alg}}$ is connected, Lemma 2.2 shows that $G_{l,L_{0},1}^{\operatorname{alg}}$ is connected. It follows from the proof of Theorem 5.4 that $G_{l,L_{0},1}^{\operatorname{alg}}=(G_{l,K,1}^{\operatorname{alg}})^{\circ}$. ∎

Theorem 5.6**.**

Let $n$ be odd. The following natural map is an isomorphism of finite groups:

[TABLE]

Proof.

This follows from Theorems 5.4 and 5.5. ∎

The main results of [BK2] concern $n$ odd. In this paper we will present a number of results that hold also for $n$ even. The following proposition holds for $(V,\psi)$ of any weight but gives essentially new information for $(V,\psi)$ of even weight.

Proposition 5.7**.**

Let $M_{0}/K$ be a field extension such that ${G_{l,M_{0}}^{\operatorname{alg}}}$ is connected. Then:

• (a)

$\mathbb{G}_{m}{\rm{Id}}_{V_{l}}\cdot({G_{l,M_{0},1}^{\operatorname{alg}}})^{\circ}\,=\,\mathbb{G}_{m}{\rm{Id}}_{V_{l}}\cdot{G_{l,M_{0},1}^{\operatorname{alg}}}\,=\,G_{l,M_{0}}^{\operatorname{alg}}$.

• (b)

${G_{l,M_{0},1}^{\operatorname{alg}}}=({G_{l,M_{0},1}^{\operatorname{alg}}})^{\circ}\,\cup\,-{\rm{Id}}_{V_{l}}\cdot({G_{l,M_{0},1}^{\operatorname{alg}}})^{\circ}$.

• (c)

$-{\rm{Id}}_{V_{l}}\in({G_{l,M_{0},1}^{\operatorname{alg}}})^{\circ}$* iff $({G_{l,M_{0},1}^{\operatorname{alg}}})^{\circ}={G_{l,M_{0},1}^{\operatorname{alg}}}$.*

• (d)

When $n$ is odd then $-{\rm{Id}}_{V_{l}}\,\in\,({G_{l,M_{0},1}^{\operatorname{alg}}})^{\circ}\,=\,{G_{l,M_{0},1}^{\operatorname{alg}}}$.

Proof.

(a) Consider a coset $g({G_{l,M_{0},1}^{\operatorname{alg}}})^{\circ}$ in $G_{l,M_{0},1}^{\operatorname{alg}}$. Applying [Hu, Section 7.4, Prop. B(b)] to the homomorphism

[TABLE]

we observe that $\mathbb{G}_{m}{\rm{Id}}_{V_{l}}\cdot({G_{l,M_{0},1}^{\operatorname{alg}}})^{\circ}$ and $\mathbb{G}_{m}{\rm{Id}}_{V_{l}}\cdot g({G_{l,M_{0},1}^{\operatorname{alg}}})^{\circ}$ are closed in $G_{l,M_{0}}^{\operatorname{alg}}$. They are also of the same dimension as $G_{l,M_{0}}^{\operatorname{alg}}$ because of the following exact sequence:

[TABLE]

Because $G_{l,M_{0}}^{\operatorname{alg}}$ is irreducible by assumption, we obtain

[TABLE]

(b) From (5.7) there are $\alpha,\beta\in\mathbb{G}_{m}$ and $g_{1},g_{2}\in({G_{l,M_{0},1}^{\operatorname{alg}}})^{\circ}$ such that:

[TABLE]

Applying $\chi$ to (5.8) we obtain $\alpha^{2}=\beta^{2}$. This implies that $\pm{\rm{Id}}_{V_{l}}\cdot g_{1}=gg_{2}$. Hence $g({G_{l,M_{0},1}^{\operatorname{alg}}})^{\circ}=({G_{l,M_{0},1}^{\operatorname{alg}}})^{\circ}$ or $g({G_{l,M_{0},1}^{\operatorname{alg}}})^{\circ}=-{\rm{Id}}_{V_{l}}({G_{l,M_{0},1}^{\operatorname{alg}}})^{\circ}$.

(c) This follows immediately from (b).

(d) For such an $M_{0}$, in the case of $n$ odd, the group ${G_{l,M_{0},1}^{\operatorname{alg}}}$ is also connected. The proof is the same as the proof of Theorem 5.5. Hence (d) follows immediately from (c). ∎

Proposition 5.8**.**

Let $M_{0}/K$ be a field extension such that ${G_{l,M_{0}}^{\operatorname{alg}}}$ is connected. Then we have the following statements.

• (a)

${G_{l,M_{0}}^{\operatorname{alg}}}\,=\,\mathbb{G}_{m}{\rm{Id}}_{V_{l}}\cdot\overline{\rho_{l}(G_{M_{0}})_{1}}$.

• (b)

$\overline{\rho_{l}(G_{M_{0}})_{1}}\,=\,({G_{l,M_{0},1}^{\operatorname{alg}}})^{\circ}$*   or   $\overline{\rho_{l}(G_{M_{0}})_{1}}\,=\,{G_{l,M_{0},1}^{\operatorname{alg}}}$.*

• (c)

$-{\rm{Id}}_{V_{l}}\in\overline{\rho_{l}(G_{M_{0}})_{1}}$*    iff    $\overline{\rho_{l}(G_{M_{0}})_{1}}={G_{l,M_{0},1}^{\operatorname{alg}}}$.*

• (d)

If $-{\rm{Id}}_{V_{l}}\notin\overline{\rho_{l}(G_{M_{0}})_{1}}$    then    $\overline{\rho_{l}(G_{M_{0}})_{1}}=({G_{l,M_{0},1}^{\operatorname{alg}}})^{\circ}$.

• (e)

When $n$ is odd then $-{\rm{Id}}_{V_{l}}\,\in\,\overline{\rho_{l}(G_{M_{0}})_{1}}\,=\,({G_{l,M_{0},1}^{\operatorname{alg}}})^{\circ}\,=\,{G_{l,M_{0},1}^{\operatorname{alg}}}$.

Proof.

(a) Let $H_{l}:=\rho_{l}(G_{M_{0}})\,\cap\,\,\mathbb{Q}_{l}^{\times}{\rm{Id}}_{V_{l}}$ denote the subgroup of homotheties of the group $\rho_{l}(G_{M_{0}})$. Because $\rho_{l}(G_{M_{0}})$ is compact, the restriction of $\chi$ to $\mathbb{Q}_{l}$-points is continuous for the $l$-adic topology, and $\mathbb{Z}_{l}^{\times}$ is the maximal compact subgroup of $\mathbb{Q}_{l}^{\times}$, we have $\chi(\rho_{l}(G_{M_{0}}))\subset\mathbb{Z}_{l}^{\times}$. Hence we have the following exact sequence of groups:

[TABLE]

It follows by Bogomolov’s theorem on homotheties that the group $\mathbb{Z}_{l}^{\times}/\chi(H_{l})$ is finite. Hence $[\rho_{l}(G_{M_{0}})\colon H_{l}\cdot\rho_{l}(G_{M_{0}})_{1}]\,<\,\infty$. In this way we obtain the equality

[TABLE]

for some coset representatives $g_{i}\in\rho_{l}(G_{M_{0}})$. Recall that $\overline{\rho_{l}(G_{M_{0}})_{1}}\,\subset\,G_{l,M_{0},1}^{\operatorname{alg}}$. Hence:

[TABLE]

Since the second and third terms from the left of (5.11) are closed in $G_{l,M_{0}}^{\operatorname{alg}}$ (cf. the proof of Proposition 5.7(a)), then taking the Zariski closure of the left term of (5.11) in $G_{l,M_{0}}^{\operatorname{alg}}$ we obtain

[TABLE]

The same argument as in the proof of Proposition 5.7(a) shows that all the sets $g_{i}\,(\mathbb{G}_{m}{\rm{Id}}_{V_{l}}\cdot\overline{\rho_{l}(G_{M_{0}})_{1}})$ are closed in $G_{l,M_{0},1}^{\operatorname{alg}}$ and of the same dimension as $G_{l,M_{0}}^{\operatorname{alg}}$. Because $G_{l,M_{0}}^{\operatorname{alg}}$ is irreducible the equality (5.12) implies

[TABLE]

(b) Observe that $\mathbb{G}_{m}{\rm{Id}}_{V_{l}}\,\cap\,G_{l,M_{0},1}^{\operatorname{alg}}=\{\pm 1\}$. Hence it follows from (5.13) that $\overline{\rho_{l}(G_{M_{0}})_{1}}$ has the same dimension as $G_{l,M_{0},1}^{\operatorname{alg}}$. Since $(G_{l,M_{0},1}^{\operatorname{alg}})^{\circ}$ is irreducible, the claim follows by Proposition 5.7(b).

(c) This follows by (b) and Proposition 5.7(c).

(d) This follows by (b) and (c).

(e) This follows by (b) and (c) and Proposition 5.7(d). ∎

Consider the natural continuous homomorphism

[TABLE]

Since $(G_{l,K}^{\operatorname{alg}})^{\circ}$ is open in $G_{l,K}^{\operatorname{alg}}$, we get

[TABLE]

for some finite Galois extension $K_{0}/K$.

Remark 5.9*.*

We observe that $K_{0}/K$ is the minimal extension such that $G_{l,K_{0}}^{\operatorname{alg}}=(G_{l,K}^{\operatorname{alg}})^{\circ}$. When $n$ is odd, it follows by Theorems 5.5 and 5.6 (equivalently [BK2, Prop. 4.7, Th. 4.8]) that $K_{0}/K$ is the minimal extension such that $G_{l,K_{0}}^{\operatorname{alg}}=(G_{l,K}^{\operatorname{alg}})^{\circ}$ and $G_{l,K_{0},1}^{\operatorname{alg}}=(G_{l,K,1}^{\operatorname{alg}})^{\circ}$. Obviously, Propositions 5.7 and 5.8 hold for $M_{0}=K_{0}$. In principle, $K_{0}$ may depend on $l$.

The following theorem and corollary extend [BK2, Prop. 4.7 and Theorem 4.8] to the arbitrary weight case.

Theorem 5.10**.**

Let $n$ be an arbitrary weight. Then:

• (a)

$G_{l,K_{0},1}^{\operatorname{alg}}=({G_{l,K,1}^{\operatorname{alg}}})^{\circ}\cup\,-{\rm{Id}}_{V_{l}}({G_{l,K,1}^{\operatorname{alg}}})^{\circ}$.

• (b)

The homomorphism

[TABLE]

is an epimorphism with kernel $\{({G_{l,K,1}^{\operatorname{alg}}})^{\circ},\,\,\,-{\rm{Id}}_{V_{l}}({G_{l,K,1}^{\operatorname{alg}}})^{\circ}\}$.

Proof.

(a) By the definition of $K_{0}$ we have $({G_{l,K}^{\operatorname{alg}}})^{\circ}={G_{l,K_{0}}^{\operatorname{alg}}}$ (see Remark 5.9). It follows by Proposition 5.7(b) and by (5.2) that

[TABLE]

(b) This follows by (a) and (5.1). ∎

Corollary 5.11**.**

Let $n$ be an arbitrary weight. Then $-{\rm{Id}}_{V_{l}}\in({G_{l,K,1}^{\operatorname{alg}}})^{\circ}$ if and only if the following conditions hold:

• (1)

$G_{l,K_{0},1}^{\operatorname{alg}}=({G_{l,K,1}^{\operatorname{alg}}})^{\circ}$;

• (2)

$i_{CC}\colon\pi_{0}(G_{l,K,1}^{\operatorname{alg}})\quad{\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}}\quad\pi_{0}(G_{l,K}^{\operatorname{alg}})$.

Proof.

This follows immediately from Proposition 5.7 (c) and Theorem 5.10. ∎

Define

[TABLE]

Lemma 5.12**.**

We have the following equality

[TABLE]

Proof.

We have

[TABLE]

Definition 5.13**.**

The Serre $l$-adic parity group:

[TABLE]

is the component group of $G_{l,K,1}^{\operatorname{alg},\,0}=G_{l,K_{0},1}^{\operatorname{alg}}$.

Proposition 5.14**.**

If $n$ is odd or $({G_{l,K,1}^{\operatorname{alg}}})^{\circ}=C_{D}(\operatorname{Iso}_{(V_{l},\psi_{l})})$ then $\operatorname{P}_{\rm{S}}(V_{l},\psi_{l})$ is trivial.

Proof.

This follows from Remark 5.9, [BK2, Remark 6.3] and the following observation:

[TABLE]

Proposition 5.15**.**

If $n$ is even and ${\rm dim}_{\mathbb{Q}}\,V$ is odd then $\operatorname{P}_{\rm{S}}(V_{l},\psi_{l})$ is nontrivial.

Proof.

Because $n$ is even, $\operatorname{Iso}_{(V_{l},\psi_{l})}=\operatorname{O}_{(V_{l},\psi_{l})}$. Hence $({G_{l,K,1}^{\operatorname{alg}}})^{\circ}\subset\operatorname{Iso}_{(V_{l},\psi_{l})}\subset{\pm{\rm{Id}}_{V_{l}}}\operatorname{SL}_{V_{l}}$. It follows that $-{\rm{Id}}_{V_{l}}\notin({G_{l,K,1}^{\operatorname{alg}}})^{\circ}$ because ${\rm{det}}(-{\rm{Id}}_{V_{l}})=(-1)^{{\rm{dim}}_{\mathbb{Q}_{l}}\,V_{l}}=-1$ and $({G_{l,K,1}^{\operatorname{alg}}})^{\circ}$ is connected. ∎

Example 5.16**.**

Let $A$ be an abelian threefold over $K$. Let $V_{l}:=H_{et}^{2}({\overline{A}},\,\mathbb{Q}_{l})=\bigwedge^{2}\,H_{et}^{1}({\overline{A}},\,\mathbb{Q}_{l})$. Then by Example 2.15 and Proposition 5.15, the group $\operatorname{P}_{\rm{S}}(V_{l},\psi_{l})$ is nontrivial.

Example 5.17**.**

Let $A$ be an elliptic curve over $K$. Let $V_{l}:=\operatorname{Sym}^{2}\,H_{et}^{1}({\overline{A}},\,\mathbb{Q}_{l})$. Then by Example 2.16 and Proposition 5.15, the group $\operatorname{P}_{\rm{S}}(V_{l},\psi_{l})$ is nontrivial.

In [BK2, Chapter 6] we proved two technical results [BK2, Lemma 6.8] and [BK2, Corollary 6.9] that hold for any weight $n$. These results imply two theorems [BK2, Theorems 6.10 and 6.11] that also hold for any weight $n$. We state these theorems below. Put

[TABLE]

Theorem 5.18**.**

Assume that the following conditions hold:

• (1)

$K_{e}\,\cap\,K(\mu_{\bar{l}}^{\otimes\,n})\,=\,K$,

• (2)

$1+l\mathbb{Z}_{l}\,\,{\rm{Id}}_{V_{l}}\,\subset\,\rho_{l}(G_{K})$.

Then all arrows in the following commutative diagram are isomorphisms:

Proof.

See [BK2, Theorem 6.10]. ∎

Theorem 5.19**.**

Assume that the following two conditions hold:

• (1)

$K_{0}\,\cap\,K(\mu_{\bar{l}}^{\otimes\,n})\,=\,K$,

• (2)

$1+l\mathbb{Z}_{l}\,\,{\rm{Id}}_{V_{l}}\,\subset\,\rho_{l}(G_{K})$.

Then all arrows in the following commutative diagram are isomorphisms:

Moreover each coset of $G_{K}/G_{K_{0}}$ has the form $\tilde{\sigma}_{1}\,G_{K_{0}}$ such that:

[TABLE]

Proof.

See [BK2, Theorem 6.11]. ∎

Theorem 5.20**.**

Let $n$ be an arbitrary weight. Assume that:

• (1)

$K_{0}\,\cap\,K(\mu_{\bar{l}}^{\otimes\,n})\,=\,K$,

• (2)

$1+l\mathbb{Z}_{l}\,\,{\rm{Id}}_{V_{l}}\,\subset\,\rho_{l}(G_{K})$,

• (3)

$-{\rm{Id}}_{V_{l}}\in({G_{l,K,1}^{\operatorname{alg}}})^{\circ}$.

Then the following equality holds:

[TABLE]

Proof.

By the assumption (3) and Theorem 5.10, we have $G_{l,K_{0},1}^{\operatorname{alg}}=(G_{l,K,1}^{\operatorname{alg}})^{\circ}$. Hence by Proposition 5.8 we obtain

[TABLE]

By assumptions (1) and (2), we obtain conditions (2) and (3) in the conclusion of Theorem 5.19. By equation (5.23) above, the right-hand side of (5.21) is the Zariski closure of the right-hand side of (5.20). Hence the left-hand side of (5.21) is the Zariski closure of the left-hand side of (5.20). ∎

Corollary 5.21**.**

Let $n$ be odd. Assume that:

• (1)

$K_{0}\,\cap\,K(\mu_{\bar{l}}^{\otimes\,n})\,=\,K$,

• (2)

$1+l\mathbb{Z}_{l}\,\,{\rm{Id}}_{V_{l}}\,\subset\,\rho_{l}(G_{K})$.

Then the following equality holds:

[TABLE]

Proof.

Because $n$ is odd, the assumption (3) of Theorem 5.20 follows by Proposition 5.7(d). ∎

Remark 5.22*.*

When $A/K$ is an abelian variety over $K$ and $\overline{A}:=A\otimes_{K}\overline{K}$, then the $\mathbb{Q}$-vector space $V_{A}=H^{1}(\overline{A}(\mathbb{C}),\mathbb{Q})$ admits a rational, polarized, pure Hodge structure $(V_{A},\psi_{A})$ of weight 1 associated with a polarization of $A$. Hence Corollary 5.21 can be applied to the $l$-adic representation of $G_{K}$ on $V_{l}=H^{1}_{et}(\overline{A},\mathbb{Q}_{l})\simeq T_{l}(A)^{\vee}$.

Remark 5.23*.*

In Theorems 5.10, 5.1, 5.19, 5.20 and Corollaries 5.11, 5.21, we can replace $K$ with a finite extension $L$ and $K_{0}$ with the corresponding $L_{0}$ (resp. $K_{e}$ with the corresponding $L_{e}$). If $K\subset L\subset K_{0}$, then $K_{0}=L_{0}$ and $K_{e}=L_{e}$.

6. Computation of the identity connected component of $G_{l,K,1}^{\operatorname{alg}}$

In this section we compare our approach with the approach of Serre for the setup of the Sato–Tate conjecture. We apply this comparison to compute the identity connected component of $G_{l,K,1}^{\operatorname{alg}}$.

Compare the $l$-adic representation $\rho_{l}$ in (4.1) and the $l$-adic representation $\widetilde{\rho}_{l}$ considered by Serre cf. [Se6, p. 111–112], given by the diagonal action of $G_{K}$ on the $\mathbb{Q}_{l}$-vector space $V_{l}\oplus\mathbb{Q}_{l}(1)$:

[TABLE]

[TABLE]

where $\chi_{c}$ is the $l$-adic cyclotomic character.

Remark 6.1*.*

In loc. cit., Serre has the summand $\mathbb{Q}_{l}(-1)$ because he works with geometric Frobenius.

Put $W_{l}:=V_{l}\oplus\mathbb{Q}_{l}(1)$. Let $\widetilde{G_{l,K}^{\operatorname{alg}}}:=\widetilde{G_{l,K}^{\operatorname{alg}}(W_{l})}$ denote the Zariski closure of $\widetilde{\rho}_{l}$ in $\operatorname{GL}_{W_{l}}$. Observe that

[TABLE]

Applying the projections onto the first and second factor of $G_{l,K}^{\operatorname{alg}}\times\mathbb{G}_{m}$ we obtain the natural morphism $\pi\colon\widetilde{G_{l,K}^{\operatorname{alg}}}\rightarrow G_{l,K}^{\operatorname{alg}}$ and the morphism $N\colon\widetilde{G_{l,K}^{\operatorname{alg}}}\rightarrow\mathbb{G}_{m}$ defined by Serre [Se6, p. 111–112].

Consider the following diagram.

The morphisms $(x\mapsto x^{-n})\circ N$ and $\chi\circ\pi$ in the right square of Diagram 6.1 are equal when restricted to the dense subset $\widetilde{\rho}_{l}(G_{K})$ of $\widetilde{G_{l,K}^{\operatorname{alg}}}$ because of (4.2). By an argument as in the proof of [Ha, Chap. I, Lemma 4.1] these morphisms are equal on $\widetilde{G_{l,K}^{\operatorname{alg}}}$. Hence the right square of Diagram 6.1 commutes. The map $\pi_{1}$ in this diagram is defined by $\pi$ and clearly makes the left square also commute.

The middle vertical arrow of Diagram 6.1 is an epimorphism because its image contains the dense subset $\rho_{l}(G_{K})$ of $G_{l,K}^{\operatorname{alg}}$. But the image of $\pi$ is closed in $G_{l,K}^{\operatorname{alg}}$ by [Hu, Chap. 2, Sec. 7.4, Proposition B(b)].

By the definition of $N$ and (6.2), the kernel of $\pi$ in Diagram 6.1 is contained in ${\rm{Id}}_{V_{l}}\times\mathbb{G}_{m}$. Hence the homomorphism $N$ restricted to kernel of $\pi$ is a monomorphism. By the commutativity of Diagram 6.1, the kernel of $\pi$ injects into $\mu_{n}$. Hence $\pi_{1}$ is a monomorphism. In addition $\dim\widetilde{G_{l,K}^{\operatorname{alg}}}=\dim G_{l,K}^{\operatorname{alg}}$ and consequently $\dim\widetilde{G_{l,K,1}^{\operatorname{alg}}}=\dim G_{l,K,1}^{\operatorname{alg}}$. This shows that

[TABLE]

Define

[TABLE]

Lemma 6.2**.**

The following exact sequence splits:

[TABLE]

Moreover $\widetilde{G_{l,K,1}^{\operatorname{alg},\,0}}$ is connected, i.e. $\widetilde{G_{l,K,1}^{\operatorname{alg},\,0}}=(\widetilde{G_{l,K,1}^{\operatorname{alg}}})^{\circ}$.

Proof.

Let $\lambda\subset\mathcal{O}_{K}$ be a prime over $l$. Let

[TABLE]

denote the restrictions of $\rho_{l}$ and $\widetilde{\rho}_{l}$ to $G_{K_{\lambda}}\subset G_{K}$. The representations $\rho_{l}^{\lambda}$ and $\widetilde{\rho}_{l}^{\lambda}$ are Hodge–Tate. Let $G_{l,K_{\lambda}}^{\operatorname{alg}}$ (resp. $\widetilde{G_{l,K_{\lambda}}^{\operatorname{alg}}}$) be the Zariski closure of $\rho_{l}(G_{K_{\lambda}})$ in $\operatorname{GL}_{V_{l}}$ (resp. $\widetilde{\rho}_{l}(G_{K_{\lambda}})$ in $\operatorname{GL}_{W_{l}}$). There are cocharacters:

[TABLE]

such that for each $x\in\mathbb{C}_{l}^{\times}=\mathbb{G}_{m}(\mathbb{C}_{l})$ the cocharacter $h_{\lambda}(x)$ (resp. $\widetilde{h}_{\lambda}(x))$ acts on the subspace $V(i)$ (resp. $W(i)$) of weight $i$ of $V_{\mathbb{C}_{l}}$ (resp. $W_{\mathbb{C}_{l}}$) via multiplication by $x^{i}$. Because $\mathbb{Q}_{l}(1)$ is a $\mathbb{Q}_{l}$-rational Hodge–Tate module of weight $1$, we obtain $\widetilde{h}_{\lambda}(x)=(h_{\lambda}(x),x)$. By [Se4, p. 158–159] the cocharacters $h_{\lambda},\widetilde{h}_{\lambda}$ have smaller targets as follows:

[TABLE]

Observe that $G_{l,K_{\lambda}}^{\operatorname{alg}}\subset(G_{l,K}^{\operatorname{alg}})_{K_{\lambda}}$ and $\widetilde{G_{l,K_{\lambda}}^{\operatorname{alg}}}\subset(\widetilde{G_{l,K}^{\operatorname{alg}}})_{K_{\lambda}}$. Enlarging targets we obtain cocharacters:

[TABLE]

such that $\widetilde{h}(x)=(h(x),x)$. By construction $N\circ\widetilde{h}(x)=x$. The cocharacter $\widetilde{h}$ is a splitting of $N$.

Because $\mathbb{G}_{m}/\mathbb{C}_{l}$ is connected, there is the following exact sequence split by $\widetilde{h}$:

[TABLE]

Because $\mathbb{C}_{l}\simeq\mathbb{C}$ the following exact sequence splits:

[TABLE]

By Lemma 2.2 the algebraic group $\widetilde{G_{l,K,1}^{\operatorname{alg},\,0}}$ is connected. ∎

Theorem 6.3**.**

*(Serre)

There is the following isomorphism:*

[TABLE]

Proof.

This follows from [Se6, p. 113], but we instead recall the proof of [BK1, Th. 3.3] or [BK2, Th. 4.6]. Consider the following commutative diagram.

In Diagram 6.2, all rows and the middle column are obviously exact. The left column is exact by Lemma 6.2. By a chase in Diagram 6.3, it follows that the right column is exact. ∎

Consider the following commutative diagram.

In Diagram 6.3, the bottom row is the exact sequence of Theorem 5.4. By the discussion above concerning Diagram 6.1 we observe that the map $\overline{\pi}$ in Diagram 6.3 is an epimorphism and the map $\overline{\pi}_{1}$ in Diagram 6.3 is a monomorphism. Theorem 5.10 and a chase in Diagram 6.3 show that

[TABLE]

where $-{\rm{Id}}_{V_{l}}(G_{l,K,1}^{\operatorname{alg}})^{\circ}$ denotes the coset of $-{\rm{Id}}_{V_{l}}$ in the quotient group $\pi_{0}(G_{l,K,1}^{\operatorname{alg}})=G_{l,K,1}^{\operatorname{alg}}/(G_{l,K,1}^{\operatorname{alg}})^{\circ}$.

Consider the following commutative diagram with exact rows.

The equality (6.6) and a chase in Diagram 6.4 show that

[TABLE]

so in particular $[G_{l,K,1}^{\operatorname{alg}}:\,\widetilde{G_{l,K,1}^{\operatorname{alg}}}]\leq 2$ and $\widetilde{G_{l,K,1}^{\operatorname{alg}}}\,\triangleleft\,G_{l,K,1}^{\operatorname{alg}}$.

Remark 6.4*.*

Observe that the equality (6.7) continues to hold with $K$ replaced by any finite extension of $K$.

Corollary 6.5**.**

$(G_{l,K,1}^{\operatorname{alg}})^{\circ}=\widetilde{G_{l,K_{0},1}^{\operatorname{alg}}}$.

Proof.

It follows from Diagram 6.1, with $K$ replaced by $K_{0}$, that $\widetilde{G_{l,K_{0},1}^{\operatorname{alg}}}$ is a closed subgroup of $G_{l,K,1}^{\operatorname{alg}}$. Hence the equality (6.7) with $K_{0}$ in place of $K$ shows that $\widetilde{G_{l,K_{0},1}^{\operatorname{alg}}}$ is also an open subgroup of $G_{l,K,1}^{\operatorname{alg}}$. Hence comparing the equality (6.7) with $K_{0}$ in place of $K$, the equality in Proposition 5.7(b) with $K_{0}$ in place of $M_{0}$, and the equality (6.3), we obtain $(\widetilde{G_{l,K_{0},1}^{\operatorname{alg}}})^{\circ}=\widetilde{G_{l,K_{0},1}^{\operatorname{alg}}}$. Moreover by (6.3) again we obtain $(G_{l,K,1}^{\operatorname{alg}})^{\circ}\,=\,(\widetilde{G_{l,K_{0},1}^{\operatorname{alg}}})^{\circ}$. ∎

Remark 6.6*.*

The group $\widetilde{G_{l,K_{0}}^{\operatorname{alg}}}$ is connected (replace $K$ with $K_{0}$ in Diagram 6.3) and

[TABLE]

By Corollary 6.5, $\widetilde{G_{l,K_{0},1}^{\operatorname{alg}}}$ is the identity connected component of $\widetilde{G_{l,K,1}^{\operatorname{alg}}}$. Hence

[TABLE]

Lemma 6.7**.**

When $\operatorname{P}_{\rm{S}}(V_{l},\psi_{l})$ is trivial then

[TABLE]

In particular, when $n$ is odd then (6.10) holds.

Proof.

The equality (6.10) holds by Theorem 5.10(a) and the equalities (6.3) and (6.7). When $n$ is odd, the parity group $\operatorname{P}_{\rm{S}}(V_{l},\psi_{l})$ is trivial by Proposition 5.14. ∎

Remark 6.8*.*

Observe that when $\operatorname{P}_{\rm{S}}(V_{l},\psi_{l})$ is trivial, the equality (6.10) follows immediately from Diagrams 6.3 and 6.4.

Remark 6.9*.*

Under the assumptions and notation of Theorem 5.19 we have

[TABLE]

Theorem 6.10**.**

Under the assumptions and notation of Theorem 5.19 we have:

[TABLE]

where $i$ runs over $\{0\}$ if $\rho_{l}(\tilde{\sigma}_{1})\in\widetilde{G_{l,K,1}^{\operatorname{alg}}}$ and over $\{0,1\}$ if $\rho_{l}(\tilde{\sigma}_{1})\notin\widetilde{G_{l,K,1}^{\operatorname{alg}}}$.

Proof.

If $-{\rm{Id}}_{V_{l}}\in\widetilde{G_{l,K_{0},1}^{\operatorname{alg}}}$, then by the equality (6.7) for $K$ and $K_{0}$ we obtain $\widetilde{G_{l,K_{0},1}^{\operatorname{alg}}}=G_{l,K_{0},1}^{\operatorname{alg}}$ and $\widetilde{G_{l,K,1}^{\operatorname{alg}}}=G_{l,K,1}^{\operatorname{alg}}$. Hence (6.12) is just (5.21).

If $-{\rm{Id}}_{V_{l}}\in\widetilde{G_{l,K,1}^{\operatorname{alg}}}\,-\,\widetilde{G_{l,K_{0},1}^{\operatorname{alg}}}$, then by the equality (6.7) for $K$ and $K_{0}$ we obtain $\widetilde{G_{l,K,1}^{\operatorname{alg}}}=G_{l,K,1}^{\operatorname{alg}}$ and $\widetilde{G_{l,K_{0},1}^{\operatorname{alg}}}\not=G_{l,K_{0},1}^{\operatorname{alg}}$. In addition the cosets $(-{\rm{Id}}_{V_{l}})^{i}\,\rho_{l}(\tilde{\sigma}_{1})\,\widetilde{G_{l,K_{0},1}^{\operatorname{alg}}}$, indexed by cosets $\tilde{\sigma}_{1}G_{K_{0}}$ (cf. (5.21)) and numbers $i=0,1$ are pairwise distinct because $-{\rm{Id}}_{V_{l}}\in G_{l,K_{0},1}^{\operatorname{alg}}$. Hence (6.13) holds.

If $-{\rm{Id}}_{V_{l}}\notin\widetilde{G_{l,K,1}^{\operatorname{alg}}}$ then $-{\rm{Id}}_{V_{l}}\notin\widetilde{G_{l,K_{0},1}^{\operatorname{alg}}}$. By equality (6.7) for $K$ and $K_{0}$ we obtain $\widetilde{G_{l,K,1}^{\operatorname{alg}}}\,\cap\,\,G_{l,K_{0},1}^{\operatorname{alg}}=\widetilde{G_{l,K_{0},1}^{\operatorname{alg}}}$. As observed above, the cosets $(-{\rm{Id}}_{V_{l}})^{i}\,\rho_{l}(\tilde{\sigma}_{1})\,\widetilde{G_{l,K_{0},1}^{\operatorname{alg}}}$, indexed by cosets $\tilde{\sigma}_{1}G_{K_{0}}$ (cf. (5.21)) and numbers $i=0,1$ are pairwise distinct because $-{\rm{Id}}_{V_{l}}\in G_{l,K_{0},1}^{\operatorname{alg}}$. Hence (6.14) holds. ∎

Lemma 6.11**.**

All arrows in the following commutative diagram are isomorphisms:

Proof.

The right vertical arrow is an epimorphism because of Remark 6.6: it is the same as the right vertical arrow $\overline{\pi}$ in Diagram 6.3. It follows directly from the definition of Zariski closure that the right horizontal arrows $\widetilde{\operatorname{Zar}}_{K_{0}/K}$ and ${\operatorname{Zar}}_{K_{0}/K}$ are epimorphisms. The left horizontal arrows are clearly epimorphisms. Hence the middle vertical arrow is also an epimorphism. By [BK2, (6.4)] the composition of the bottom horizontal arrows is an isomorphism; note that [BK2, (6.4)] holds for arbitrary weight $n$. Now a chase in Diagram 6.5 shows that all arrows are isomorphisms. ∎

Let $\tilde{\sigma}\in G_{K}$ be a lift of $\sigma\in\operatorname{Gal}(K_{0}/K)$. The coset $\tilde{\sigma}\,G_{K_{0}}$ does not depend on the lift. The Zariski closure of $\widetilde{\rho}_{l}(\tilde{\sigma}\,G_{K_{0}})=\widetilde{\rho}_{l}(\tilde{\sigma})\,\widetilde{\rho}_{l}(G_{K_{0}})$ in $\operatorname{GL}_{W_{l}}$ is $\widetilde{\rho}_{l}(\tilde{\sigma})\,\widetilde{G_{l,K_{0}}^{\operatorname{alg}}}$. It follows immediately from Lemma 6.5 that:

[TABLE]

Put:

[TABLE]

Let $K\subseteq L\subset\overline{F}$ be a tower of extensions with $L/K$ finite. Consider the following commutative diagram, in which the left and middle vertical arrows are monomorphisms.

Observe that $\widetilde{G_{l,K,1}^{\operatorname{alg}}}\,\cap\,\widetilde{G_{l,L}^{\operatorname{alg}}}\,=\,\widetilde{G_{l,L,1}^{\operatorname{alg}}}$. If $L/K$ is Galois, then it follows from the Diagram 6.6 that there is a monomorphism:

[TABLE]

Theorem 6.12**.**

Let $v_{0}$ be a prime in $\mathcal{O}_{K_{0}}$ such that $v_{0}|l$. Let $K_{v_{0}}$ be the completion of $K_{0}$ at $v_{0}$. Assume that $\chi_{c}(G_{K})=\chi_{c}(G_{{K_{v_{0}}}})$. Then all arrows in the following commutative diagram are isomorphisms:

Moreover each coset of $G_{K}/G_{K_{0}}$ has the form $\tilde{\sigma}_{1}\,G_{K_{0}}$ such that:

• (1)

$\widetilde{\rho}_{l}(\tilde{\sigma}_{1})\in\widetilde{\rho}_{l}(G_{K})_{1}$,

• (2)

$\widetilde{\rho}_{l}(G_{K})_{1}\,\,=\,\,{\bigsqcup}_{\tilde{\sigma}_{1}G_{K_{0}}}\,\,\widetilde{\rho}_{l}(\tilde{\sigma}_{1})\,\widetilde{\rho}_{l}(G_{K_{0}})_{1}$,

• (3)

$\widetilde{G_{l,K,1}^{\operatorname{alg}}}\,\,=\,\,{\bigsqcup}_{\tilde{\sigma}_{1}G_{K_{0}}}\,\,\widetilde{\rho}_{l}(\tilde{\sigma}_{1})\widetilde{\,G_{l,K_{0},1}^{\operatorname{alg}}}$.

Proof.

The right vertical arrow $\tilde{i}_{CC}$ is an isomorphism by Theorem 6.3. The top horizontal arrow $\widetilde{\operatorname{Zar}}_{K_{0}/K}$ is an isomorphism by Lemma 6.5. The left vertical arrow $\tilde{j}_{K_{0}/K}$ is a monomorphism by (6.17). Take any $\sigma\in\operatorname{Gal}(K_{0}/K)=G_{K}/G_{K_{0}}$. Let $\tilde{\sigma}\in G_{K}$ be a lift of $\sigma$. By (6.1), (6.2), and the definition of $N$ we obtain the following equality for any $\tau\in G_{K}$:

[TABLE]

By assumption we can choose $\gamma\in G_{K_{v_{0}}}\subset G_{K_{0}}\subset G_{K}$ such that $\chi_{c}(\tilde{\sigma}\,\gamma^{-1})=1$. So $\widetilde{\rho}_{l}(\tilde{\sigma}\gamma^{-1})\in\widetilde{\rho}_{l}(G_{K})_{1}$ by (6.18). Hence we obtain

[TABLE]

Therefore $\tilde{j}_{K_{0}/K}$ is an epimorphism, hence an isomorphism. This proves that the bottom horizontal arrow $\widetilde{\operatorname{Zar}}_{K_{0}/K,\,1}$ is also an isomorphism. Put $\tilde{\sigma}_{1}:=\tilde{\sigma}\gamma^{-1}$. We have $\tilde{\sigma}_{1}\,G_{K_{0}}=\tilde{\sigma}\,G_{K_{0}}$ and $\widetilde{\rho}_{l}(\tilde{\sigma}_{1})\in\widetilde{\rho}_{l}(G_{K})_{1}$, so claim (1) follows. Because $\tilde{j}_{K_{0}/K}$ and $\widetilde{\operatorname{Zar}}_{K_{0}/K,\,1}$ are isomorphisms, claims (2) and (3) follow by applying (6.15) and (6.16). ∎

Remark 6.13*.*

Because $\widetilde{G_{l,K_{0}}^{\operatorname{alg}}}$ and $\widetilde{G_{l,K_{0},1}^{\operatorname{alg}}}$ are connected, everywhere in §6 we can replace $K$ with a finite extension $L$ and $K_{0}$ with the corresponding $L_{0}$. If $K\subset L\subset K_{0}$, then $K_{0}=L_{0}$. After such replacement, all results in §6 hold for the base field $L$ with $K_{0}$ replaced by $L_{0}$.

Remark 6.14*.*

Consider the following commutative diagram:

In Diagram 6.8, the rows and the middle column are exact, the left vertical arrow is an embedding, and the kernel of $\pi$ injects into the kernel of the right vertical arrow, cf. the discussion following Diagram 6.1.

Corollary 6.15**.**

Assume $n$ and $l-1$ are coprime. Then:

• (a)

$\widetilde{\rho}_{l}(G_{K})=\rho_{l}(G_{K})$,

• (b)

$\widetilde{\rho}_{l}(G_{K})_{1}=\rho_{l}(G_{K})_{1}$,

• (c)

$\widetilde{G_{l,K}^{\operatorname{alg}}}=G_{l,K}^{\operatorname{alg}}$,

• (d)

$\widetilde{G_{l,K,1}^{\operatorname{alg}}}=G_{l,K,1}^{\operatorname{alg}}$.

Proof.

By assumption the right vertical arrow in Diagram 6.8 is an isomorphism. This implies that all vertical arrows in this diagram are isomorphisms. This proves (1) and (2). The equalities (3) and (4) are proven in the same way based on the discussion following Diagram 6.1. ∎

Corollary 6.16**.**

Assume that:

• (1)

$K_{0}\,\cap\,K(\mu_{\bar{l}}^{\otimes\,n})\,=\,K$,

• (2)

$1+l\mathbb{Z}_{l}\,\,{\rm{Id}}_{V_{l}}\,\subset\,\rho_{l}(G_{K})$,

• (3)

$-{\rm{Id}}_{V_{l}}\in({G_{l,K,1}^{\operatorname{alg}}})^{\circ}$,

• (4)

$n$* and $l-1$ are coprime.*

Then the following equality holds:

[TABLE]

Proof.

This follows from Theorem 5.20 and Corollary 6.15. ∎

7. Algebraic Sato–Tate conjecture for families of $l$-adic representations

Let $A$ be an abelian variety over $K$ and $(V_{A},\psi_{A})$ be the Hodge structure on $H^{1}(A(\mathbb{C}),\mathbb{Q})$ associated with a polarization of $A$.

Theorem 7.1**.**

(Deligne [D1, I, Prop. 6.2], Piatetski-Shapiro [P-S], Borovoi [Bor]; see also [Se2, §4.1]) For any prime number $l$,

[TABLE]

The classical Mumford–Tate conjecture for $A/K$ states:

Conjecture 7.2**.**

(Mumford–Tate) For any prime number $l$,

[TABLE]

We can formulate the Mumford–Tate conjecture for families of $l$-adic representations associated with rational polarized Hodge structures satisfying the conditions (D1), (D2) of §2 and conditions (R1)–(R4) of §4 as follows.

Conjecture 7.3**.**

(Mumford–Tate) For any prime number $l$,

[TABLE]

Remark 7.4*.*

It is not known whether the analogue of the inclusion (7.1) holds for more general classes of $(V,\psi)$ beyond $(V_{A},\psi_{A})$. In any given case, if we have the following inclusion:

[TABLE]

we obtain the following commutative diagram with all horizontal arrows closed immersions and all columns exact.

It then follows immediately from Diagram 7.1, Proposition 2.1(a), (b), and Proposition 5.7(a), (b) that (7.4) is equivalent to the inclusion

[TABLE]

and Mumford–Tate Conjecture 7.3 is equivalent to the equality

[TABLE]

In [BK2, Conjectures 5.1 and 5.9], we formulated the algebraic Sato–Tate and Sato–Tate conjectures for families of strictly compatible $l$-adic representations of Hodge–Tate type associated with pure Hodge structures. In this paper we state the algebraic Sato–Tate and Sato–Tate conjectures imposing conditions (D1), (D2) and (R1)–(R4).

Conjecture 7.5**.**

(Algebraic Sato–Tate conjecture; cf. [BK2, Conjecture 5.1])

• (a)

For every finite extension $K/F$ there exist a natural-in-$K$* reductive algebraic group $\operatorname{AST}_{K}(V,\psi)\subset\operatorname{Iso}_{(V,\psi)}$ over $\mathbb{Q}$ and a natural-in-$K$ monomorphism of group schemes for every $l$:*

[TABLE]

In addition the natural embedding $G_{l,K,1}^{\operatorname{alg}}\subset\operatorname{Iso}_{(V_{l},\psi_{l})}$ factors through $\operatorname{ast}_{l,K}$ and the natural embedding $\operatorname{AST}_{K}(V,\psi)_{\mathbb{Q}_{l}}\subset\operatorname{Iso}_{(V_{l},\psi_{l})}$.

• (b)

The map (7.7) is an isomorphism:

[TABLE]

Remark 7.6*.*

The requirement that $\operatorname{AST}_{K}(V,\psi)$ and (7.7) are natural in $K$ means that for any finite extension $L/K$, there is a natural monomorphism of groups schemes $\operatorname{AST}_{K}(V,\psi)\hookrightarrow\operatorname{AST}_{L}(V,\psi)$ cf. [BK2, Remark 5.3].

Remark 7.7*.*

In [BK2, Conjecture 5.1] we had the phrase “for every l” in the wrong place. This is corrected in Conjecture 7.5 above.

Definition 7.8**.**

The group $\operatorname{AST}_{K}(V,\psi)$ is called the algebraic Sato–Tate group associated with the family of representations $(\rho_{l})_{l}$. Any maximal compact subgroup of $\operatorname{AST}_{K}(V,\psi)(\mathbb{C})$ is called the Sato–Tate group and denoted $\operatorname{ST}_{K}(V,\psi)$; recall that any two such subgroups are conjugate [Kn, §VII.2].

Proposition 7.9**.**

Assume that Algebraic Sato–Tate Conjecture 7.5 holds for $(V,\psi)$. Then there are natural isomorphisms

[TABLE]

Proof.

Given our assumptions, this follows by an argument similar to the proof of [FKRS, Lemma 2.8]. ∎

Remark 7.10*.*

Fixing $l$ and $K$ and assuming that $\operatorname{ast}_{l,K}$ is an isomorphism we can prove (7.9) for these $l$ and $K$, cf. the proof of [FKRS, Lemma 2.8].

Remark 7.11*.*

Assume that Algebraic Sato–Tate Conjecture 7.5 holds for $(V,\psi)$. Then obviously the Sato–Tate group $\operatorname{ST}_{K}(V,\psi)$ is independent of $l$. Take a prime $v$ in $\mathcal{O}_{K}$ and take a Frobenius element $\operatorname{fr}_{v}$ in $G_{K}$. Following [Se6, §8.3.3] (cf. [FKRS, Def. 2.9]) one can make the following construction. Let $s_{v}$ be the semisimple part in $\operatorname{SL}_{V}(\mathbb{C})$ of the following normalized Frobenius element:

[TABLE]

since the family $(\rho_{l})$ is strictly compatible, the conjugacy class ${\rm{conj}}(s_{v})$ in $\operatorname{SL}_{V}(\mathbb{C})$ is independent of $l$. By [Hu, Theorem 15.3 (c) p. 99], the semisimple part of $q_{v}^{-\frac{n}{2}}\rho_{l}(\operatorname{fr}_{v})$ considered in $\operatorname{Iso}_{(V,\psi)}(\mathbb{C})$ and in $\operatorname{AST}_{K}(V,\psi)(\mathbb{C})$ is again $s_{v}$, but its conjugacy classes ${\rm{conj}}(s_{v})$ in $\operatorname{Iso}_{(V,\psi)}(\mathbb{C})$ and ${\rm{conj}}(s_{v})$ in $\operatorname{AST}_{K}(V,\psi)(\mathbb{C})$ might depend on $l$. Obviously ${\rm{conj}}(s_{v})\subset\operatorname{AST}_{K}(V,\psi)(\mathbb{C})$ is independent of the choice of a Frobenius element $\operatorname{fr}_{v}$ over $v$ and contains the semisimple parts of all the elements of ${\rm{conj}}(q_{v}^{-\frac{n}{2}}\rho_{l}(\operatorname{fr}_{v}))$ in $\operatorname{AST}_{K}(V,\psi)(\mathbb{C})$. Moreover, the elements in ${\rm{conj}}(s_{v})$ have eigenvalues of complex absolute value $1$ by our assumptions, so there is some conjugate of $s_{v}$ contained in $\operatorname{ST}_{K}(V,\psi)$. This allows us to make sense of the following conjecture.

Conjecture 7.12**.**

(Sato–Tate conjecture) The conjugacy classes ${\rm{conj}}(s_{v})$ in $\operatorname{ST}_{K}(V,\psi)$ are equidistributed in ${\rm{conj}}(\operatorname{ST}_{K}(V,\psi))$ with respect to the measure induced by the Haar measure of $\operatorname{ST}_{K}(V,\psi)$.

Remark 7.13*.*

In [BK2, Remark 5.8] we claimed that $s_{v}$ is independent of $l$. That claim was too optimistic; this is still not known in general. As pointed out in Remark 7.11 above, we only know that ${\rm{conj}}(s_{v})$ in $\operatorname{SL}_{V}(\mathbb{C})$ is independent of $l$; by contrast, ${\rm{conj}}(s_{v})$ in $\operatorname{Iso}_{(V,\psi)}(\mathbb{C})$ and ${\rm{conj}}(s_{v})$ in $\operatorname{AST}_{K}(V,\psi)(\mathbb{C})$ might depend on $l$.

Theorem 7.14**.**

Assume Conjecture 7.5${(\rm{a})}$ holds for $(V,\psi)$. Assume in addition that for some $l$, the group $\operatorname{P}_{\rm{S}}(V_{l},\psi_{l})$ is trivial and the maps $\operatorname{ast}_{l,K}$ and $\operatorname{ast}_{l,K_{0}}$ are isomorphisms. Let $L/K_{0}$ be a finite Galois extension. Then:

• (a)

$\operatorname{AST}_{K_{0}}(V,\psi)=\operatorname{AST}_{K}(V,\psi)^{\circ}$,

• (b)

$\operatorname{ST}_{K_{0}}(V,\psi)=\operatorname{ST}_{K}(V,\psi)^{\circ}$* up to conjugation in $\operatorname{AST}_{K}(V,\psi)(\mathbb{C})$,*

• (c)

$\operatorname{AST}_{K_{0}}(V,\psi)=\operatorname{AST}_{L}(V,\psi)$,

• (d)

$\operatorname{ST}_{K_{0}}(V,\psi)=\operatorname{ST}_{L}(V,\psi)$* up to conjugation in $\operatorname{AST}_{K_{0}}(V,\psi)(\mathbb{C})$.*

Proof.

The proof is similar to the proof of [BK2, Prop. 6.4]. However for arbitrary weight, we need to assume that $\operatorname{P}_{\rm{S}}(V_{l},\psi_{l})$ is trivial; we then apply Proposition 7.9 to relate the Sato–Tate groups to the parity group $\operatorname{P}_{\rm{S}}(V_{l},\psi_{l})$. ∎

Proposition 7.15**.**

Assume that Conjecture 7.5 holds for $(V,\psi)$ and that the following conditions hold for some $l$:

• (1)

$K_{0}\,\cap\,K(\mu_{\bar{l}}^{\otimes\,n})\,=\,K$,

• (2)

$1+l\mathbb{Z}_{l}\,\,{\rm{Id}}_{V_{l}}\,\subset\,\rho_{l}(G_{K})$.

Then every subgroup of $\operatorname{ST}_{K}(V,\psi)$ containing $\operatorname{ST}_{K_{0}}(V,\psi)$ is of the form $\operatorname{ST}_{L}(V,\psi)$, for a unique field subextension $K\subset L\subset K_{0}$.

Proof.

By the definition of $K_{0}$ (cf. [BK2, (6.4)]), Proposition 7.9, Theorem 5.19, and the bottom row of Diagram 6.5, there are natural isomorphisms:

[TABLE]

By the fundamental theorem of Galois theory, any subgroup of $\operatorname{ST}_{K}(V,\psi)$ containing $\operatorname{ST}_{K_{0}}(V,\psi)$ corresponds via (7.10) to the subgroup $G(K_{0}/L)\subseteq G(K_{0}/K)$ for a unique subextension $K\subset L\subset K_{0}$. Hence working with the base field $L$ instead of $K$ (see Remark 5.23) we obtain again by Proposition 7.9, Theorem 5.19, and the bottom row of Diagram 6.5 the following natural isomorphisms:

[TABLE]

naturally compatible with the isomorphisms (7.10). ∎

Proposition 7.16**.**

Suppose that for some $(V,\psi)$, Conjecture 7.5 holds for $K$ and $K_{0}$ and the group $\operatorname{P}_{\rm{S}}(V_{l},\psi_{l})$ is trivial. Then the field $K_{0}$ is independent of $l$.

Proof.

Cf. the proof of [BK2, Prop. 6.5]. ∎

The following Theorem is proven in [BK2, Theorem 6.12]. The assumption that $\operatorname{ast}_{l,K_{0}}$ is an isomorphism is missing in loc. cit. so we add it here.

Theorem 7.17**.**

Assume that Conjecture 7.5 ${\rm{(a)}}$ holds for $(V,\psi)$ and the following conditions hold for some $l$:

• (1)

$K_{0}\,\cap\,K(\mu_{\bar{l}}^{\otimes\,n})\,=\,K$;

• (2)

$1+l\mathbb{Z}_{l}\,\,{\rm{Id}}_{V_{l}}\,\subset\,\rho_{l}(G_{K})$;

• (3)

$\operatorname{ast}_{l,K}$* and $\operatorname{ast}_{l,K_{0}}$ are isomorphisms.*

Then:

[TABLE]

where $\tilde{\sigma}_{1}$ runs over a set of coset representatives of $G_{K_{0}}$ in $G_{K}$ such that ${\rho}_{l}(\tilde{\sigma}_{1})\in{\rho}_{l}(G_{K})_{1}$.

Corollary 7.18**.**

Consider field extensions $K\subset K_{2}\subset K_{1}\subset K_{0}$. Under the assumptions of Theorem 7.17, we have:

[TABLE]

where $\tilde{\sigma}_{1}$ runs over a set of coset representatives of $G_{K_{1}}$ in $G_{K_{2}}$ such that ${\rho}_{l}(\tilde{\sigma}_{1})\in{\rho}_{l}(G_{K_{2}})_{1}$.

Proof.

This follows from Theorem 7.17 and Remark 5.23. ∎

To be consistent with Serre’s approach, the algebraic Sato–Tate conjecture should have now the following form.

Conjecture 7.19**.**

• (a)

For every finite extension $K/F$, there exist a natural-in-$K$* reductive algebraic group $\widetilde{\operatorname{AST}_{K}}(V,\psi)\subset\operatorname{Iso}_{(V,\psi)}$ over $\mathbb{Q}$ and a natural-in-$K$ monomorphism of group schemes for every $l$:*

[TABLE]

In addition the natural embedding $\widetilde{G_{l,K,1}^{\operatorname{alg}}}\subset\operatorname{Iso}_{(V_{l},\psi_{l})}$ factors through $\widetilde{\operatorname{ast}}_{l,K}$ and the natural embedding $\widetilde{\operatorname{AST}_{K}}(V,\psi)_{\mathbb{Q}_{l}}\subset\operatorname{Iso}_{(V_{l},\psi_{l})}$.

• (b)

The map (7.16) is an isomorphism:

[TABLE]

Definition 7.20**.**

The group $\widetilde{\operatorname{AST}_{K}}(V,\psi)$ is called the algebraic Sato–Tate group associated with the family of representations $(\widetilde{\rho}_{l})_{l}$. Any maximal compact subgroup of $\widetilde{\operatorname{AST}_{K}}(V,\psi)(\mathbb{C})$ is called the Sato–Tate group and denoted $\widetilde{\operatorname{ST}_{K}}(V,\psi)$.

Proposition 7.21**.**

Assume that Algebraic Sato–Tate Conjecture 7.19 holds for $(V,\psi)$. Then there are natural isomorphisms

[TABLE]

Proof.

This follows by the same proof as Proposition 7.9. ∎

Remark 7.22*.*

Fixing $l$ and $K$ and assuming that $\widetilde{\operatorname{ast}}_{l,K}$ is an isomorphism, we can prove (7.18) for these $l$ and $K$ cf. the proof of [FKRS, Lemma 2.8].

Theorem 7.23**.**

Assume Conjecture 7.19 ${(\rm{a})}$ holds for $(V,\psi)$. Assume in addition that for some $l$, the maps $\operatorname{ast}_{l,K}$ and $\operatorname{ast}_{l,K_{0}}$ are isomorphisms. Let $L/K_{0}$ be a finite Galois extension. Then:

• (a)

$\widetilde{\operatorname{AST}_{K_{0}}}(V,\psi)=\widetilde{\operatorname{AST}_{K}}(V,\psi)^{\circ}$,

• (b)

$\widetilde{\operatorname{ST}_{K_{0}}}(V,\psi)=\widetilde{\operatorname{ST}_{K}}(V,\psi)^{\circ}$* up to conjugation in $\widetilde{\operatorname{AST}_{K}}(V,\psi)(\mathbb{C})$,*

• (c)

$\widetilde{\operatorname{AST}_{K_{0}}}(V,\psi)=\widetilde{\operatorname{AST}_{L}}(V,\psi)$,

• (d)

$\widetilde{\operatorname{ST}_{K_{0}}}(V,\psi)=\widetilde{\operatorname{ST}_{L}}(V,\psi)$* up to conjugation in $\widetilde{\operatorname{AST}_{K_{0}}}(V,\psi)(\mathbb{C})$.*

Proof.

Upon Corollary 6.5 and the assumptions we observe that $\widetilde{\operatorname{AST}_{K_{0}}}(V,\psi)$ is connected. Hence by applying Proposition 7.21, the proof is similar to the proof of [BK2, Prop. 6.4]. ∎

Proposition 7.24**.**

Assume that Conjecture 7.19 holds for $(V,\psi)$. For some $l$, for a prime $v_{0}\,|\,l$ in $\mathcal{O}_{K_{0}}$, assume that $\chi_{c}(G_{K})=\chi_{c}(G_{{K_{v_{0}}}})$. Then every subgroup of $\widetilde{\operatorname{ST}_{K}}(V,\psi)$ containing $\widetilde{\operatorname{ST}_{K_{0}}}(V,\psi)$ has the form $\widetilde{\operatorname{ST}_{L}}(V,\psi)$ for a unique field subextension $K\subset L\subset K_{0}$.

Proof.

By Lemma 6.5, Theorem 6.12, and Proposition 7.21, there are natural isomorphisms:

[TABLE]

By the fundamental theorem of Galois theory, any subgroup of $\widetilde{\operatorname{ST}_{K}}(V,\psi)$ containing $\widetilde{\operatorname{ST}_{K_{0}}}(V,\psi)$ corresponds via (7.19) to the subgroup $G(K_{0}/L)\subseteq G(K_{0}/K)$ for a unique subextension $K\subset L\subset K_{0}$. Hence working with the base field $L$ instead of $K$ (see Remark 6.13) we obtain again by Lemma 6.5, Theorem 6.12, and Proposition 7.21 the natural isomorphisms:

[TABLE]

which are naturally compatible with the isomorphisms (7.19). ∎

Proposition 7.25**.**

Suppose that for some $(V,\psi)$, Conjecture 7.5 holds for $K$ and $K_{0}$ and the group $\operatorname{P}_{\rm{S}}(V_{l},\psi_{l})$ is trivial. Then the field $K_{0}$ is independent of $l$.

Proof.

Cf. the proof of [BK2, Prop. 6.5]. ∎

Theorem 7.26**.**

Assume that Conjecture 7.19(a)* holds for $(V,\psi)$. For some $l$, for a prime $v_{0}\,|\,l$ in $\mathcal{O}_{K_{0}}$, assume that $\chi_{c}(G_{K})=\chi_{c}(G_{{K_{v_{0}}}})$ and the maps $\widetilde{\operatorname{ast}}_{l,K}$ and $\widetilde{\operatorname{ast}}_{l,K_{0}}$ are isomorphisms. Then:*

[TABLE]

where $\tilde{\sigma}_{1}$ runs over a set of coset representatives of $G_{K_{0}}$ in $G_{K}$ such that $\widetilde{\rho}_{l}(\tilde{\sigma}_{1})\in\widetilde{\rho}_{l}(G_{K})_{1}$.

Proof.

From Theorem 6.12 we have:

[TABLE]

Because the maps $\widetilde{\operatorname{ast}}_{l,K}$ and $\widetilde{\operatorname{ast}}_{l,K_{0}}$ are isomorphisms by assumption, it follows by Proposition 7.21 and Theorem 7.23 that the equality (7.21) holds. Now taking $\mathbb{C}$-valued points in (7.21) and restricting to maximal compact subgroups proves (7.22). ∎

Corollary 7.27**.**

Consider field extensions $K\subset K_{2}\subset K_{1}\subset K_{0}$. Under the assumptions of Theorem 7.26, we have:

[TABLE]

where $\tilde{\sigma}_{1}$ runs over a set of coset representatives of $G_{K_{1}}$ in $G_{K_{2}}$ such that $\widetilde{\rho}_{l}(\tilde{\sigma}_{1})\in\widetilde{\rho}_{l}(G_{K_{2}})_{1}$.

Proof.

This follows from Theorem 7.26 and Remark 6.13. ∎

Lemma 7.28**.**

For $i=1,2$, let $V_{i}\subset W$ be an inclusion of vector spaces over a field $L$. Let $M/L$ be a field extension. If $V_{1}\otimes_{L}M\subset V_{2}\otimes_{L}M$ in $W\otimes_{L}M$, then $V_{1}\subset V_{2}$. In particular if $V_{1}\otimes_{L}M=V_{2}\otimes_{L}M\subset W\otimes_{L}M$, then $V_{1}=V_{2}$.

Proof.

It holds because $V_{1}\otimes_{L}M+V_{2}\otimes_{L}M=(V_{1}+V_{2})\otimes_{L}M$ in $W\otimes_{L}M$ and $L\hookrightarrow M$ is faithfully flat. ∎

Corollary 7.29**.**

Let $X$ and $Y$ be closed subschemes of $\mathbb{A}^{n}_{L}$ for some field $L$. Let $M/L$ be a field extension. If $X\times_{{\rm{Spec}}\,L}{\rm{Spec}}\,M\,\,\subset\,\,Y\times_{{\rm{Spec}}\,L}{\rm{Spec}}\,M$, then $X\subset Y$.

Proof.

Let $I(X)$ (resp. $I(Y)$) be the ideal in $L[t_{1},\dots,t_{n}]$ defining $X\subset\mathbb{A}^{n}_{L}$ (resp. $Y\subset\mathbb{A}^{n}_{L}$). By assumption $I(Y)\otimes_{L}\,M\subset I(X)\otimes_{L}\,M$. Hence by Lemma 7.28 $I(Y)\subset I(X)$. So $X\subset Y\subset\mathbb{A}^{n}_{L}$. ∎

Corollary 7.30**.**

Let $X={\rm{Spec}}\,A$ and $Y={\rm{Spec}}\,B$ be affine schemes over $L$. Let $M/L$ be a field extension. Let $f,g\colon X\rightarrow Y$ be two morphisms such that the morphisms $f\times{\rm{Id}},\,g\times{\rm{Id}}\colon X\times_{{\rm{Spec}}\,L}{\rm{Spec}}\,M\,\,\rightarrow\,\,Y\times_{{\rm{Spec}}\,L}{\rm{Spec}}\,M$ are equal. Then $f=g$.

Proof.

Let $f^{\sharp},g^{\sharp}\colon B\rightarrow A$ be the $L$-algebra homomorphisms corresponding to $f$ and $g$, respectively. By assumption, the $M$-algebra homomorphisms $f^{\sharp}\otimes_{L}\,1,\,g^{\sharp}\otimes_{L}\,1\colon B\otimes_{L}\,M\rightarrow A\otimes_{L}\,M$ are equal. We observe that $f^{\sharp}\otimes_{L}\,1-g^{\sharp}\otimes_{L}1=(f^{\sharp}-g^{\sharp})\otimes_{L}\,1=0$ as $M$-vector space homomorphisms. Let $W:={\rm{ker}}(f^{\sharp}-g^{\sharp})$. Then $W\otimes_{L}\,M=B\otimes_{L}\,M$. Hence by Lemma 7.28 it follows that $W=B$, so $f^{\sharp}=g^{\sharp}$. Therefore $f=g$. ∎

Corollary 7.31**.**

Assume Conjectures 7.5 and 7.19. Then:

• (a)

$\operatorname{AST}_{K}(V,\psi)^{\circ}=\operatorname{AST}_{K_{0}}(V,\psi)^{\circ}=\widetilde{\operatorname{AST}_{K_{0}}}(V,\psi)$,

• (b)

$\operatorname{ST}_{K}(V,\psi)^{\circ}=\operatorname{ST}_{K_{0}}(V,\psi)^{\circ}=\widetilde{\operatorname{ST}_{K_{0}}}(V,\psi)$* up to conjugation in $\operatorname{AST}_{K}(V,\psi)(\mathbb{C})$.*

Proof.

To prove (a) it is enough to prove $\operatorname{AST}_{K_{0}}(V,\psi)^{\circ}=\widetilde{\operatorname{AST}_{K_{0}}}(V,\psi)$. By [Hu, p. 218] the algebraic group $\operatorname{AST}_{K_{0}}(V,\psi)^{\circ}$ is defined over $\mathbb{Q}$. Consider the $\mathbb{Q}$-vector spaces $V_{1}:={\rm{Lie}}(\operatorname{AST}_{K_{0}}(V,\psi)^{\circ})$, $V_{2}:={\rm{Lie}}(\widetilde{\operatorname{AST}_{K_{0}}}(V,\psi))$, $W:={\rm{Lie}}(\operatorname{Iso}_{(V,\psi)})$. Obviously $V_{i}\subset W$ for $i=1,2$. It follows by (7.8), (7.17), and Corollary 6.5 that $V_{1}\otimes_{\mathbb{Q}}\mathbb{Q}_{l}=V_{2}\otimes_{\mathbb{Q}}\mathbb{Q}_{l}\subset W\otimes_{\mathbb{Q}}\mathbb{Q}_{l}$. By Lemma 7.28 we have $V_{1}=V_{2}$. In the end by [Hu, Theorem on p. 87] we obtain $\operatorname{AST}_{K_{0}}(V,\psi)^{\circ}=\widetilde{\operatorname{AST}_{K_{0}}}(V,\psi)$. Hence (a) holds; this in turn implies (b). ∎

Remark 7.32*.*

Under Conjecture 7.19, the Sato–Tate group $\widetilde{\operatorname{ST}_{K}}(V,\psi)$ is independent of $l$. The normalization of $\widetilde{\rho}_{l}(\operatorname{fr}_{v})$ (cf. [Se6, p. 113]) gives the following normalized Frobenius element:

[TABLE]

The monomorphism $\pi_{1}$ in Diagrams 6.1 and 6.4 sends $(q_{v}^{-\frac{n}{2}}{\rm{Id}}_{V_{l}},\,q_{v}^{-1}{\rm{Id}}_{\mathbb{Q}_{l}(1)})\,\widetilde{\rho}_{l}(\operatorname{fr}_{v})$ onto $q_{v}^{-\frac{n}{2}}{\rho}_{l}(\operatorname{fr}_{v})$. Let $\tilde{s}_{v}$ denote the semisimple part in $\operatorname{SL}_{V}(\mathbb{C})$ of the element $(q_{v}^{-\frac{n}{2}}{\rm{Id}}_{V_{l}},\,q_{v}^{-1}{\rm{Id}}_{\mathbb{Q}_{l}(1)})\,\widetilde{\rho}_{l}(\operatorname{fr}_{v})$. Observe that $\tilde{s}_{v}=(s_{v},\,q_{v}^{-1}{\rm{Id}}_{\mathbb{Q}_{l}(1)})$. The family $(\widetilde{\rho}_{l})$ is also strictly compatible. Hence the conjugacy class ${\rm{conj}}(\tilde{s}_{v})$ in $\operatorname{SL}_{V}(\mathbb{C})$ is independent of $l$. Based on [Hu, Theorem 15.3 (c) p. 99], the semisimple part $\tilde{s}_{v}$ considered in $\operatorname{Iso}_{(V,\psi)}(\mathbb{C})$ and in $\widetilde{\operatorname{AST}_{K}}(V,\psi)(\mathbb{C})$ is again $\tilde{s}_{v}$, but its conjugacy classes ${\rm{conj}}(\tilde{s}_{v})$ in $\operatorname{Iso}_{(V,\psi)}(\mathbb{C})$ and ${\rm{conj}}(\tilde{s}_{v})$ in $\widetilde{\operatorname{AST}_{K}}(V,\psi)(\mathbb{C})$ might depend on $l$. Obviously ${\rm{conj}}(\tilde{s}_{v})\subset\widetilde{\operatorname{AST}_{K}}(V,\psi)(\mathbb{C})$ is independent of the choice of a Frobenius element $\operatorname{fr}_{v}$ over $v$ and contains the semisimple parts of all the elements of ${\rm{conj}}((q_{v}^{-\frac{n}{2}}{\rm{Id}}_{V_{l}},\,q_{v}^{-1}{\rm{Id}}_{\mathbb{Q}_{l}(1)})\,\widetilde{\rho}_{l}(\operatorname{fr}_{v}))$ in $\widetilde{\operatorname{AST}_{K}}(V,\psi)(\mathbb{C})$. The elements in ${\rm{conj}}(\tilde{s}_{v})$ have eigenvalues of complex absolute value $1$ by our assumptions, so there is some conjugate of $\tilde{s}_{v}$ contained in $\widetilde{\operatorname{ST}_{K}}(V,\psi)$.

Conjecture 7.33**.**

(Sato–Tate conjecture) The conjugacy classes ${\rm{conj}}(\tilde{s}_{v})$ in $\widetilde{\operatorname{ST}_{K}}(V,\psi)$ are equidistributed in ${\rm{conj}}(\widetilde{\operatorname{ST}_{K}}(V,\psi))$ with respect to the measure induced by the Haar measure of $\widetilde{\operatorname{ST}_{K}}(V,\psi)$.

Proposition 7.34**.**

• (a)

Under Conjecture 7.5 the algebraic Sato–Tate group $\operatorname{AST}_{K}(V,\psi)$ is uniquely defined.

• (b)

Under Conjecture 7.19 the algebraic Sato–Tate group $\widetilde{\operatorname{AST}_{K}}(V,\psi)$ is uniquely defined.

• (c)

Under Conjectures 7.5 and 7.19 we have $\widetilde{\operatorname{AST}_{K}}(V,\psi)\subset\operatorname{AST}_{K}(V,\psi)$. Hence $\widetilde{\operatorname{ST}_{K}}(V,\psi)\subset\operatorname{ST}_{K}(V,\psi)$ up to conjugation in $\operatorname{AST}_{K}(V,\psi)(\mathbb{C})$.

Proof.

Claims (a) and (b) follow from Corollary 7.29. Claim (c) follows from (6.7) and Corollary 7.29. ∎

Proposition 7.35**.**

Conjecture 7.19 implies Conjecture 7.5.

Proof.

Assume that Conjecture 7.19 holds. Then we put

[TABLE]

It is obvious that $\operatorname{AST}_{K}(V,\psi)$ is an algebraic group over $\mathbb{Q}$ and Conjecture 7.5 holds by (6.7). ∎

Remark 7.36*.*

Assume Conjecture 7.5. For $K_{0}$ in place of $K$ we can put

[TABLE]

which is an algebraic group defined over $\mathbb{Q}$ cf. [Hu, p. 218]. Hence for the base field $K_{0}$, Conjecture 7.19 holds with $\widetilde{\operatorname{AST}_{K_{0}}}(V,\psi)=\operatorname{AST}_{K_{0}}(V,\psi)^{\circ}$ by Corollary 6.5. By (6.7) with $K_{0}$ in place of $K$, we have:

[TABLE]

Moreover if $-{\rm{Id}}_{V_{l}}\notin\widetilde{G_{l,K_{0},1}^{\operatorname{alg}}}$ then $-{\rm{Id}}_{V}\notin\widetilde{\operatorname{AST}_{K_{0}}}(V,\psi)$ and

[TABLE]

Remark 7.37*.*

Assume Conjecture 7.5. If $-{\rm{Id}}_{V_{l}}\in\widetilde{G_{l,K,1}^{\operatorname{alg}}}$ then $\widetilde{G_{l,K,1}^{\operatorname{alg}}}=G_{l,K,1}^{\operatorname{alg}}$ by the equality (6.7). Putting:

[TABLE]

we observe that Conjecture 7.19 holds in this case.

If $-{\rm{Id}}_{V_{l}}\notin\widetilde{G_{l,K,1}^{\operatorname{alg}}}$, the equality (6.7) has the form

[TABLE]

Hence $G_{l,K,1}^{\operatorname{alg}}/\{\pm{\rm{Id}}_{V_{l}}\}=\widetilde{G_{l,K,1}^{\operatorname{alg}}}$. Since a quotient map of algebraic groups is uniquely determined by its kernel [Mi2, p.101], it follows that

[TABLE]

By [Mi2, Theorems 1.72, 3.34, 5.14, Corollary 5.18], the group scheme $\operatorname{AST}_{K}(V,\psi)/\{\pm{\rm{Id}}_{V}\}$ is affine and defined over $\mathbb{Q}$. We put

[TABLE]

Consider the following commutative diagram with exact rows.

By (7.29) and Diagram 7.2, we obtain the following natural-in-$K$ isomorphism:

[TABLE]

By [Mi2, Proposition 19.13] and (7.30), the group scheme $\widetilde{\operatorname{AST}_{K}}(V,\psi)$ is reductive because $\widetilde{G_{l,K,1}^{\operatorname{alg}}}$ is reductive. Nevertheless we cannot fully establish Conjecture 7.19 in this case because it is not clear how to prove that $\widetilde{\operatorname{AST}_{K}}(V,\psi)\subset\operatorname{Iso}_{(V,\psi)}$ over $\mathbb{Q}$. Proposition 8.17 and Proposition 17.6 suggest that the condition $\widetilde{\operatorname{AST}_{K}}(V,\psi)\subset\operatorname{Iso}_{(V,\psi)}$ over $\mathbb{Q}$ is naturally expected in the setting of this paper.

Remark 7.38*.*

Assume Algebraic Sato–Tate Conjecture 7.19. By Remark 7.32 there is a one-to-one correspondence between normalized Frobenius elements in $\operatorname{AST}_{K}(V,\psi)(\mathbb{C})$ and normalized Frobenius elements in $\widetilde{\operatorname{AST}_{K}}(V,\psi)(\mathbb{C})$. Similarly there is a one-to-one correspondence between conjugacy classes of normalized Frobenius elements in $\operatorname{AST}_{K}(V,\psi)(\mathbb{C})$ and conjugacy classes of normalized Frobenius elements in $\widetilde{\operatorname{AST}_{K}}(V,\psi)(\mathbb{C})$ because $\widetilde{\operatorname{AST}_{K}}(V,\psi)\,\triangleleft\,\operatorname{AST}_{K}(V,\psi)$ and $-{\rm{Id}}_{V}$ is in the center of $\operatorname{AST}_{K}(V,\psi)$. Hence if $-{\rm{Id}}_{V}\notin\,\widetilde{\operatorname{AST}_{K}}(V,\psi)$ then there are no conjugacy classes of normalized Frobenius elements in $-{\rm{Id}}_{V}\,\widetilde{\operatorname{AST}_{K}}(V,\psi)(\mathbb{C})$.

Corollary 7.39**.**

Assume Algebraic Sato–Tate Conjecture 7.19. Then

[TABLE]

Moreover the following conditions are equivalent:

• (1)

$-{{\rm Id}}_{V}\in\widetilde{\operatorname{AST}_{K}}(V,\psi)$,

• (2)

$-{{\rm Id}}_{V}\in\widetilde{\operatorname{ST}_{K}}(V,\psi)$,

• (3)

$\operatorname{AST}_{K}(V,\psi)=\widetilde{\operatorname{AST}_{K}}(V,\psi)$,

• (4)

$\operatorname{ST}_{K}(V,\psi)=\widetilde{\operatorname{ST}_{K}}(V,\psi)$.

Proof.

The equality (7.31) is immediate from the proof of Proposition 7.35. The equality (7.32) follows from (7.31). Now the equivalence of conditions (1)–(4) is obvious. ∎

Remark 7.40*.*

For a fixed base field $K$, the field $K_{0}$ depends in general on $l$ (see Remark 5.9). This observation motivates the following definition.

Definition 7.41**.**

Under Algebraic Sato–Tate Conjecture 7.19 the Sato–Tate parity group is defined as follows:

[TABLE]

Remark 7.42*.*

By Corollary 7.39 the group $\operatorname{P}_{\rm{ST}}(V,\psi)$ is either trivial or isomorphic to $\mathbb{Z}/2\mathbb{Z}$. It follows from Lemma 6.7 that the natural map $\operatorname{P}(V_{l},\psi_{l})\rightarrow\operatorname{P}_{\rm{ST}}(V,\psi)$ is an epimorphism.

Theorem 7.43**.**

Assume Algebraic Sato–Tate Conjecture 7.19.

• (a)

If $\operatorname{P}_{\rm{ST}}(V,\psi)$ is nontrivial, i.e. $-{{\rm Id}}_{V}\notin\widetilde{\operatorname{ST}_{K}}(V,\psi)$, then Sato–Tate Conjecture 7.12 does not hold.

• (b)

If $\operatorname{P}_{\rm{ST}}(V,\psi)$ is trivial, i.e. $-{{\rm Id}}_{V}\in\widetilde{\operatorname{ST}_{K}}(V,\psi)$, then Sato–Tate Conjectures 7.12 and 7.33 are equivalent.

Proof.

If $-{{\rm Id}}_{V}\notin\widetilde{\operatorname{ST}_{K}}(V,\psi)$ then

[TABLE]

Let $\mu$ be the probabilistic Haar measure on $\operatorname{ST}_{K}(V,\psi)$. Consider the following character:

[TABLE]

Observe that

[TABLE]

On the other hand, let $m\in\mathbb{N}$ and $N_{K}\,(m):=\,\#\{v\colon Nv\,\leq\,m\}$. By Remark 7.38 we obtain

[TABLE]

Hence the sequence ${\rm{conj}}(s_{v})$ is not equidistributed in the conjugacy classes of $\operatorname{ST}_{K}(V,\psi)$, so (a) holds.

By the same token, if Sato–Tate Conjecture 7.12 holds, then conditions (1)–(4) of Corollary 7.39 hold, so in particular condition (4) holds. In this case $\mu=\tilde{\mu}$ and by Remark 7.38, ${\rm{conj}}(\tilde{s}_{v})={\rm{conj}}(s_{v})$ in $\widetilde{\operatorname{ST}_{K}}(V,\psi)=\operatorname{ST}_{K}(V,\psi)$. Hence ${\rm{conj}}(\tilde{s}_{v})$ is equidistributed in $\widetilde{\operatorname{ST}_{K}}(V,\psi)$. Hence (b) holds. ∎

8. Tate conjecture and Algebraic Sato–Tate conjecture for families of $l$-adic representations

The framework of this section provides a wider perspective on Algebraic Sato–Tate Conjectures 7.5 and 7.19. The following two Conjectures 8.1 and 8.2 are analogues of the Tate conjecture for motives. For a special case of these conjectures see [LP]. Throughout this section (except in Proposition 8.4) we assume only the weaker form of Conjectures 8.1 and 8.2, namely Conjectures 8.1(a) and 8.2(a). For a wide range of examples where Conjectures 8.1(a) and 8.2(a) are known to hold, see Remark 8.3.

Let $(V,\psi)$ be a rational, polarized, pure Hodge structure of weight $n$. Let

[TABLE]

be a family of $l$-adic representations (4.1) satisfying the conditions (D1), (D2), (DR1), (DR2), (R1)–(R4) of §2, §3, §4.

Conjecture 8.1**.**

• (a)

For every finite extension $K/F$, there exist a natural-in-$K$* reductive group scheme $\mathcal{G}_{K}^{\operatorname{alg}}:=\mathcal{G}_{K}^{\operatorname{alg}}(V,\psi)\subset\operatorname{GIso}_{(V,\psi)}$ over $\mathbb{Q}$ and a natural-in-$K$ monomorphism of group schemes for every $l$:*

[TABLE]

In addition the natural embedding $G_{l,K}^{\operatorname{alg}}\subset\operatorname{GIso}_{(V_{l},\psi_{l})}$ factors through ${\rm{t}}_{l,K}$ and the natural embedding ${\mathcal{G}_{K}^{\operatorname{alg}}}_{{\mathbb{Q}_{l}}}\subset\operatorname{GIso}_{(V_{l},\psi_{l})}$.

• (b)

The homomorphism (8.1) is an isomorphism:

[TABLE]

Consider the rational Hodge structure $W:=V\oplus\mathbb{Q}(1)$. Then $W_{l}=V_{l}\oplus\mathbb{Q}_{l}(1)$. Consider the family of $l$-adic representations $(\widetilde{\rho}_{l})$ associated with $W$ (see (6.1)):

[TABLE]

Conjecture 8.2**.**

• (a)

For every finite extension $K/F$, there exists a natural-in-$K$* reductive group scheme $\widetilde{\mathcal{G}_{K}^{\operatorname{alg}}}:=\widetilde{\mathcal{G}_{K}^{\operatorname{alg}}}(W)\subset\mathcal{G}_{K}^{\operatorname{alg}}\times{\mathbb{G}}_{m}\subset\operatorname{GL}_{W}$ over $\mathbb{Q}$ such that projections onto the factors of $\mathcal{G}_{K}^{\operatorname{alg}}\times{\mathbb{G}}_{m}$ give epimorphisms:*

[TABLE]

and there is a natural monomorphism:

[TABLE]

In addition, the natural embedding $\widetilde{G_{l,K}^{\operatorname{alg}}}\subset\operatorname{GL}_{W_{l}}$ factors through $\widetilde{{\rm{t}}}_{l,K}$ and the natural embedding $\widetilde{{\mathcal{G}_{K}^{\operatorname{alg}}}}_{{\mathbb{Q}_{l}}}\subset\operatorname{GL}_{W_{l}}$.

• (b)

The homomorphism (8.3) is an isomorphism:

[TABLE]

Remark 8.3*.*

We recall three examples of families of $l$-adic representations associated with rational, pure, polarized Hodge structures for which Conjectures 8.1(a) and 8.2(a) are satisfied.

The first family comes from the category of polarized realizations which is a full subcategory of Jannsen’s category of realizations. We define and describe in detail the category of polarized realizations in §9. See Proposition 9.13 for the families of $l$-adic representations satisfying Conjectures 8.1(a) and 8.2(a).

The second and third families, described in §17, come from motives in the Deligne motivic category for absolute Hodge cycles and motives in the André motivic category for motivated cycles.

Proposition 8.4**.**

• (a)

Under Conjecture 8.1 the group scheme $\mathcal{G}_{K}^{\operatorname{alg}}$ is uniquely defined.

• (b)

Under Conjecture 8.2 the group scheme $\widetilde{\mathcal{G}_{K}^{\operatorname{alg}}}$ is uniquely defined.

Proof.

Claims (a) and (b) follow from Corollary 7.29. ∎

Definition 8.5**.**

Let

[TABLE]

denote the restriction of the character $\chi$ of the polarization $\psi$ to $\mathcal{G}_{K}^{\operatorname{alg}}$ (cf. 2.1).

Definition 8.6**.**

Put:

[TABLE]

Lemma 8.7**.**

Under Conjectures 8.1(a) and 8.2(a) the following Diagram 8.1 commutes and the map $\pi_{1}$ is a monomorphism.

Proof.

We already know that the morphisms $N$, $\pi$, and $x\mapsto x^{-n}$ are epimorphisms. We next check that the morphism $\chi$ in Diagram 8.1 is an epimorphism. Extending base to $\mathbb{Q}_{l}$ in Diagram 8.1, we observe that under Conjectures 8.1(a) and 8.2 a), the right square of Diagram 6.1 embeds into the right square of the resulting diagram (cf. Diagram 8.2 below). Hence $\chi_{{}_{\mathbb{Q}_{l}}}\colon{\mathcal{G}_{K}^{\operatorname{alg}}}_{{\mathbb{Q}_{l}}}\rightarrow{\mathbb{G}_{m}}$ is an epimorphism. Therefore $\chi$ in Diagram 8.1 is also an epimorphism by [Wa, p. 106, first corollary].

In addition, the right square of Diagram 8.1 commutes by Corollary 7.30 applied to the field extension $\mathbb{Q}_{l}/\mathbb{Q}$. The left square of Diagram 8.1 obviously commutes with the morphism $\pi_{1}$ induced by $\pi$. Because of $\widetilde{\mathcal{G}_{K}^{\operatorname{alg}}}\subset\mathcal{G}_{K}^{\operatorname{alg}}\times{\mathbb{G}}_{m}$ and the definition of $N$, the kernel of $\pi$ is contained in ${\rm{Id}}_{V}\times\mathbb{G}_{m}$. Hence the homomorphism $N$ restricted to the kernel of $\pi$ is a monomorphism. By the commutativity of Diagram 8.1, the kernel of $\pi$ injects into $\mu_{n}$. Hence $\pi_{1}$ is a monomorphism. ∎

In the framework of Conjectures 8.1(a) and 8.2(a), it is natural to make the following definition.

Definition 8.8**.**

Put:

[TABLE]

Every maximal compact subgroup of $\operatorname{AST}_{K}(V,\psi)(\mathbb{C})$ (resp. $\widetilde{\operatorname{AST}_{K}}(V,\psi)(\mathbb{C}))$ will be called a Sato–Tate group and denoted $\operatorname{ST}_{K}(V,\psi)$ (resp. $\widetilde{\operatorname{ST}_{K}}(V,\psi)$).

Consider the following double cube commutative Diagram 8.2. In light of Definition 8.8 it is natural to denote by $\widetilde{{\rm{ast}}}_{l,K}$ and ${\rm{ast}}_{l,K}$ the horizontal arrows in the left wall of this diagram.

Assume Conjectures 8.1(a) and 8.2(a). The back wall of Diagram 8.2 is Diagram 6.1. The horizontal sequences of homomorphisms in the front wall of Diagram 8.2 are exact because the kernel of a group scheme homomorphism is invariant under base change (Remark 8.9 below) and any field extension (in particular $\mathbb{Q}_{l}/\mathbb{Q}$) is flat. For the same reasons, the arrows in these horizontal sequences in the left cube are monomorphisms and in the right cube are epimorphisms.

Remark 8.9*.*

Let $\phi\colon G\rightarrow G^{\prime}$ be a homomorphism of $S$-group schemes. Let $S^{\prime}$ be an $S$-scheme. Define the $S$-homomorphism $e^{\prime}\colon S\rightarrow G^{\prime}$ to be the unit section. Recall that ${{\rm Ker}}\,\phi=G\times_{G^{\prime}}S$. For any scheme $Z$ and any $Z$-schemes $X,Y$ denote as usual $X(Y)_{Z}:={\rm{Hom}}_{Z}(Y,\,X)$. Then for any $S^{\prime}$-group scheme $T$ in the category of group schemes over $S$:

[TABLE]

By uniqueness of the representing object we obtain:

[TABLE]

Remark 8.10*.*

Under Conjectures 8.1(a) and 8.2(a), the homomorphisms ${\rm{ast}}_{l,K}$ and $\widetilde{{\rm{ast}}}_{l,K}$ in the left cube in Diagram 8.2 satisfy all conditions of Algebraic Sato–Tate Conjectures 7.5(a) and 7.19(a). It is immediate from Diagram 8.2 that ${\rm{ast}}_{l,K}$ (resp. $\widetilde{{\rm{ast}}}_{l,K}$) is an isomorphism iff ${\rm{t}}_{l,K}$ (resp. $\widetilde{{{\rm{t}}}}_{l,K}$) is an isomorphism.

Remark 8.10 leads directly to the following corollary.

Corollary 8.11**.**

Assume Conjectures 8.1(a) and 8.2(a). Then Algebraic Sato–Tate Conjecture 7.5 (resp. 7.19) holds if and only if Tate Conjecture 8.1 (resp. 8.2) holds.

Assume Conjectures 8.1(a) and 8.2(a). Recall from the proof of Lemma 8.7 that in Diagram 8.1, the kernel of $\pi$ injects into $\mu_{n}$ and $\pi_{1}$ is a monomorphism. Hence $\dim\widetilde{\mathcal{G}_{K}^{\operatorname{alg}}}=\dim\mathcal{G}_{K}^{\operatorname{alg}}$ and consequently $\dim\widetilde{\mathcal{G}_{K,1}^{\operatorname{alg}}}=\dim\mathcal{G}_{K,1}^{\operatorname{alg}}$. This shows that

[TABLE]

Define:

[TABLE]

Lemma 8.12**.**

The following exact sequence splits:

[TABLE]

Moreover $\widetilde{\mathcal{G}_{K,1}^{\operatorname{alg},\,0}}$ is connected, i.e.

[TABLE]

Proof.

Consider the commutative diagram in the lid of the cube Diagram 8.2. Enlarging the target of the cocharacter $\widetilde{h}$ from the proof of Lemma 6.2 we obtain the following cocharacter:

[TABLE]

It is clear from the lid of Diagram 8.2 that $\widetilde{{\rm{t}}}_{l,K}\circ\widetilde{h}$ splits $N$ in the exact sequence 8.9. Because $\mathbb{G}_{m}$ is connected, the following exact sequence splits:

[TABLE]

Now we finish the proof in a similar way as we finished the proof of Lemma 6.2. ∎

The following theorem is proven in the same way as Theorem 6.3.

Theorem 8.13**.**

There is the following isomorphism:

[TABLE]

Proof.

Consider the following commutative diagram.

In Diagram 8.3, all rows and the middle column are obviously exact. The left column is exact by Lemma 8.12. Chasing in Diagram 8.3, we observe that the right column is exact. ∎

Lemma 8.14**.**

Assume Conjectures 8.1(a) and 8.2(a). Let $n$ be odd. Then the following exact sequence splits:

[TABLE]

Moreover $\mathcal{G}_{K,1}^{\operatorname{alg},\,0}$ is connected, i.e.

[TABLE]

Proof.

Let $L_{0}$ be such that $(G_{l,K}^{\operatorname{alg}})^{\circ}=G_{l,L_{0}}^{\operatorname{alg}}$. In the proof of Theorem 5.5 we constructed the following homomorphism

[TABLE]

which splits $\chi$ in the following exact sequence:

[TABLE]

Define

[TABLE]

It is clear that $s_{0}$ is a splitting of $\chi$ in the exact sequence (8.12). Because $(\mathcal{G}_{K}^{\operatorname{alg}})^{\circ}$ is connected, Lemma 2.2 shows that $\mathcal{G}_{K,1}^{\operatorname{alg},\,0}$ is connected. In particular, (8.13) holds. ∎

Proposition 8.15**.**

Assume Conjectures 8.1(a) and 8.2(a). Then:

• (a)

$\mathbb{G}_{m}{\rm{Id}}_{V}\cdot(\mathcal{G}_{K,1}^{\operatorname{alg}})^{\circ}\,=\,\mathbb{G}_{m}{\rm{Id}}_{V}\cdot{\mathcal{G}_{K,1}^{\operatorname{alg},\,0}}\,=\,(\mathcal{G}_{K}^{\operatorname{alg}})^{\circ}$;

• (b)

${\mathcal{G}_{K,1}^{\operatorname{alg},\,0}}=(\mathcal{G}_{K,1}^{\operatorname{alg}})^{\circ}\,\cup\,-{\rm{Id}}_{V}\cdot({\mathcal{G}_{K,1}^{\operatorname{alg}}})^{\circ}$;

• (c)

$-{\rm{Id}}_{V}\in({\mathcal{G}_{K,1}^{\operatorname{alg}}})^{\circ}$* iff $({\mathcal{G}_{K,1}^{\operatorname{alg}}})^{\circ}={\mathcal{G}_{K,1}^{\operatorname{alg},\,0}}$;*

• (d)

when $n$ is odd, $-{\rm{Id}}_{V}\,\in\,({\mathcal{G}_{K,1}^{\operatorname{alg}}})^{\circ}\,=\,{\mathcal{G}_{K,1}^{\operatorname{alg},\,0}}$.

Proof.

The proof is similar to the proof of Proposition 5.7. First of all observe that $\mathbb{G}_{m}{\rm{Id}}_{V}\subset{{\mathcal{G}_{K}^{\operatorname{alg}}}}$ by Conjecture 8.1(a) and Corollary 7.29.

(a) Consider a coset $g({\mathcal{G}_{K,1}^{\operatorname{alg}}})^{\circ}$ in $\mathcal{G}_{K,1}^{\operatorname{alg},\,0}$. Applying [Hu, Section 7.4, Prop. B(b)] to the homomorphism

[TABLE]

we observe that $\mathbb{G}_{m}{\rm{Id}}_{V}\cdot({{\mathcal{G}_{K,1}^{\operatorname{alg}}}})^{\circ}$ and $\mathbb{G}_{m}{\rm{Id}}_{V}\cdot g({\mathcal{G}_{K,1}^{\operatorname{alg}}})^{\circ}$ are closed in $({\mathcal{G}_{K}^{\operatorname{alg}}})^{\circ}$.

They are also of the same dimension as ${({\mathcal{G}_{K}^{\operatorname{alg}}})^{\circ}}$ because of the following exact sequence:

[TABLE]

Because $({\mathcal{G}_{K}^{\operatorname{alg}}})^{\circ}$ is irreducible, we obtain

[TABLE]

(b) From (5.7) there are $\alpha,\beta\in\mathbb{G}_{m}$ and $g_{1},g_{2}\in({\mathcal{G}_{K,1}^{\operatorname{alg}}})^{\circ}$ such that:

[TABLE]

Applying $\chi$ to (8.16) we obtain $\alpha^{2}=\beta^{2}$. This implies that $\pm{\rm{Id}}_{V}\cdot g_{1}=gg_{2}$. Hence $g({\mathcal{G}_{K,1}^{\operatorname{alg}}})^{\circ}=({\mathcal{G}_{K,1}^{\operatorname{alg}}})^{\circ}$ or $g({\mathcal{G}_{K,1}^{\operatorname{alg}}})^{\circ}=-{\rm{Id}}_{V}({\mathcal{G}_{K,1}^{\operatorname{alg}}})^{\circ}$.

(c) This follows immediately from (b).

(d) In the case of $n$ odd, the group ${\mathcal{G}_{K,1}^{\operatorname{alg},\,0}}$ is connected by Lemma 8.14. Hence (d) follows immediately from (c). ∎

Consider the following commutative diagram.

In Diagram 8.4, the top row is the isomorphism (8.11). The map $\overline{\pi}$ in Diagram 8.4 is an epimorphism because the map $\pi$ in Conjecture 8.2(a) is assumed to be an epimorphism. Exactness of the bottom sequence follows from surjectivity of $\overline{\pi}$ and the definition of $\mathcal{G}_{K,1}^{\operatorname{alg},\,0}$. The map $\overline{\pi}_{1}$ in Diagram 8.4 is a monomorphism because the map $\pi_{1}$ in Diagram 8.1 is a monomorphism and because of (8.8). Proposition 8.15(b), equality (8.8), and a chase in Diagram 8.4 show that

[TABLE]

where $-{\rm{Id}}_{V}(\mathcal{G}_{K,1}^{\operatorname{alg}})^{\circ}$ denotes the coset of $-{\rm{Id}}_{V}$ in the quotient group $\pi_{0}(\mathcal{G}_{K,1}^{\operatorname{alg}})=\mathcal{G}_{K,1}^{\operatorname{alg}}/(\mathcal{G}_{K,1}^{\operatorname{alg}})^{\circ}$.

Consider the following commutative diagram with exact rows.

Equality (8.17) and a chase in Diagram 8.5 show that

[TABLE]

In particular $[\mathcal{G}_{K,1}^{\operatorname{alg}}:\,\widetilde{\mathcal{G}_{K,1}^{\operatorname{alg}}}]\leq 2$ and $\widetilde{\mathcal{G}_{K,1}^{\operatorname{alg}}}\,\triangleleft\,\mathcal{G}_{K,1}^{\operatorname{alg}}$.

Corollary 8.16**.**

Assume Conjectures 8.1(a) and 8.2(a). Then the following equalities hold:

• (a)

$\operatorname{AST}_{K}(V,\psi)^{\circ}\,=\,\widetilde{\operatorname{AST}_{K}}(V,\psi)^{\circ}\,=\,\widetilde{\mathcal{G}_{K,1}^{\operatorname{alg},\,0}}$.

• (b)

$\operatorname{AST}_{K}(V,\psi)\,=\,\widetilde{\operatorname{AST}_{K}}(V,\psi)\,\cup\,-{\rm{Id}}_{V}\,\widetilde{\operatorname{AST}_{K}}(V,\psi)$.

• (c)

$\operatorname{ST}_{K}(V,\psi)\,=\,\widetilde{\operatorname{ST}_{K}}(V,\psi)\,\cup\,-{\rm{Id}}_{V}\,\widetilde{\operatorname{ST}_{K}}(V,\psi)$* up to conjugation in *

$\operatorname{AST}_{K}(V,\psi)(\mathbb{C})$.

Proof.

(a) follows by (8.8) and (8.10), (b) follows by (8.18), and (c) follows by (b). ∎

Proposition 8.17**.**

Assume Conjectures 8.1(a) and 8.2(a). Then Algebraic Sato–Tate Conjecture 7.5 holds for $\operatorname{AST}_{K}(V,\psi)$ if and only Algebraic Sato–Tate Conjecture 7.19 holds for $\widetilde{\operatorname{AST}_{K}}(V,\psi)$.

Proof.

Remark 8.10 shows that Conjectures 7.5(a) and 7.19(a) hold. In particular the homomorphisms ${\rm{ast}}_{l,K}$ and $\widetilde{{\rm{ast}}}_{l,K}$ in Diagram 8.2 are monomorphisms. Hence the proposition follows by (6.7) and Corollary 8.16(b). ∎

Because of Proposition 8.17, it is enough to discuss the Algebraic Sato–Tate conjecture (in the framework of Conjectures 8.1 and 8.2) only for $\operatorname{AST}_{K}(V,\psi)$.

Corollary 8.18**.**

Assume Conjectures 8.1(a) and 8.2(a). Then the following four conjectures are equivalent:

• •

Tate Conjecture 8.1,

• •

Tate Conjecture 8.2,

• •

Algebraic Sato–Tate Conjecture 7.5,

• •

Algebraic Sato–Tate Conjecture 7.19.

Proof.

It follows by Corollary 8.11 and Proposition 8.17. ∎

Remark 8.19*.*

Under any of the four equivalent conjectures in Corollary 8.18, the group schemes $\operatorname{AST}_{K}(V,\psi)$ and $\widetilde{\operatorname{AST}_{K}}(V,\psi)$ are uniquely defined, cf. Proposition 7.34(a) and (b).

Proposition 8.20**.**

Assume Conjectures 8.1(a) and 8.2(a). Then the Sato–Tate Conjecture for $\operatorname{ST}_{K}(V,\psi)$ (Conjecture 7.12) implies the Sato–Tate Conjecture for $\widetilde{\operatorname{ST}_{K}}(V,\psi)$ (Conjecture 7.33).

Proof.

This follows by Theorem 7.43. ∎

The Sato–Tate parity group $\operatorname{P}_{\rm{ST}}(V,\psi)$ was introduced in Definition 7.41. Under Conjectures 8.1(a) and 8.2(a) and their consequences, the Definition 7.41 of $\operatorname{P}_{\rm{ST}}(V,\psi)$ is clearly valid in the framework of this section:

[TABLE]

Due to Corollary 8.16, $\operatorname{P}_{\rm{ST}}(V,\psi)$ is either trivial or isomorphic to $\mathbb{Z}/2\mathbb{Z}$.

Now we can extend Theorem 7.43 to the setup of this section.

Theorem 8.21**.**

Assume Conjectures 8.1(a) and 8.2(a).

• (a)

If $\operatorname{P}_{\rm{ST}}(V,\psi)$ is nontrivial, i.e. $-{{\rm Id}}_{V}\notin\widetilde{\operatorname{ST}_{K}}(V,\psi)$, then Sato–Tate Conjecture 7.12 does not hold.

• (b)

If $\operatorname{P}_{\rm{ST}}(V,\psi)$ is trivial, i.e. $-{{\rm Id}}_{V}\in\widetilde{\operatorname{ST}_{K}}(V,\psi)$, then Sato–Tate Conjectures 7.12 and 7.33 are equivalent.

Proof.

This follows by the proof of Theorem 7.43. ∎

Definition 8.22**.**

Assume Conjectures 8.1(a) and 8.2(a). Define the Serre parity group $\operatorname{P}_{\rm{S}}(V,\psi)$:

[TABLE]

Consider the following commutative diagram.

In Diagram 8.6, the map $\pi$ is an epimorphism by the setup of this section; consequently the map $\overline{\pi}$ is an epimorphism. By Diagram 8.1 the map $\pi$ has finite kernel. Hence the algebraic groups $\widetilde{\mathcal{G}_{K}^{\operatorname{alg}}}$ and $\mathcal{G}_{K}^{\operatorname{alg}}$ have the same dimension and the map $\pi^{\circ}$ has finite kernel. Because the image of $\pi^{\circ}$ is a closed subgroup of $(\mathcal{G}_{K}^{\operatorname{alg}})^{\circ}$ [Hu, Section 7.4, Prop. B(b)] of the same dimension as $(\mathcal{G}_{K}^{\operatorname{alg}})^{\circ}$, the image is also open. Because the group $(\mathcal{G}_{K}^{\operatorname{alg}})^{\circ}$ is connected, the map $\pi^{\circ}$ must be an epimorphism. By the snake lemma applied to Diagram 8.6 there is the following short exact sequence:

[TABLE]

Proposition 8.23**.**

There is the following short exact sequence:

[TABLE]

Proof.

By Definition 8.8 and Diagram 8.5 we have:

[TABLE]

Now the claim follows by the snake lemma applied to Diagram 8.4. ∎

9. The category of polarized realizations $R_{K}^{\rm{p}}$

Let $L\subset\mathbb{C}$ be a field. Let $w\colon\mathcal{C}\rightarrow{\rm{Vec}}_{L}$ be an exact, faithful, $\otimes$-functor from an abelian, $L$-linear, rigid, $\otimes$-category $\mathcal{C}$ to ${\rm{Vec}}_{L}$ (the category of finite-dimensional $L$-vector spaces).

Definition 9.1**.**

The group scheme $\operatorname{Aut}^{\otimes}w$ is defined as follows:

For every $L$-algebra $R$, $\operatorname{Aut}^{\otimes}w\,(R)$ consists of families $(\lambda_{X})$, $X\in{\rm{obj}}(\mathcal{C})$ such that $\lambda_{X}$ is an $R$-linear automorphism:

[TABLE]

satisfying the following conditions:

• (a)

$\lambda_{X}\otimes\lambda_{X^{\prime}}=\lambda_{X\otimes X^{\prime}}$,     for all $X,X^{\prime}\,\in\,{\rm{obj}}(\mathcal{C})$,

• (b)

$\lambda_{{\bf{1}}}\colon w({\bf{1}})\otimes R\,{\stackrel{{\scriptstyle{\rm{id}}}}{{\longrightarrow}}}\,w({\bf{1}})\otimes R$,

• (c)

the following Diagram 9.1 commutes for all $\alpha\in\operatorname{Hom}_{\mathcal{C}}(X,Y)$ and all $X,Y\in{\rm{obj}}(\mathcal{C})$.

Theorem 9.2**.**

(Tannaka Duality)* Let ${\rm{Rep}}_{L}G$ denote the category of finite dimensional, $L$-linear representations of a group scheme $G/L$.*

• (a)

${\rm{Rep}}_{L}G$* is a neutral, Tannakian category with forgetful fiber functor $w^{G}\colon{\rm{Rep}}_{L}\,G\rightarrow\,{\rm{Vec}}_{L}$ such that $\operatorname{Aut}^{\otimes}w^{G}=G$.*

• (b)

If $\mathcal{C}$ is an $L$-linear, neutral, Tannakian category with fiber functor $w\colon\mathcal{C}\rightarrow{\rm{Vec}}_{L}$, then there exist a group scheme $G/L$ satisfying $\operatorname{Aut}^{\otimes}w=G$ and an equivalence of categories $\mathcal{C}\,\,{\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}}\,\,{\rm{Rep}}_{L}G$ via which $w$ corresponds to $w^{G}$.

Proof.

See [DM, pp 128–138]. ∎

Below, up to Proposition 9.13, $K$ can be any field of characteristic 0. Jannsen [Ja1] introduced the $\mathbb{Q}$-linear, Tannakian category of mixed realizations $MR_{K}$ and its $\mathbb{Q}$-linear Tannakian full subcategory of realizations $R_{K}$. An object of $R_{K}$ is a direct sum of objects of the following form:

[TABLE]

where $\sigma$ and $\bar{\sigma}$ run over all embeddings $\sigma\colon K\hookrightarrow\mathbb{C}$,   $\bar{\sigma}\colon\overline{K}\hookrightarrow\mathbb{C}$, such that $\bar{\sigma}_{|K}=\sigma$ and $l$ runs over all prime numbers; $V_{{}_{\rm{DR}}}$ is a $K$-vector space; $V_{l}$ is a $\mathbb{Q}_{l}$-vector space with $G_{K}$-module structure; and $V_{\sigma}$ is a pure $\mathbb{Q}$-Hodge structure of weight $n$ for any $n\in\mathbb{Z}$. In addition, the vector spaces $V_{{}_{\rm{DR}}}$, $V_{l}$, and $V_{\sigma}$ are finite dimensional and there are comparison isomorphisms:

[TABLE]

such that $g\circ I_{l,\bar{\sigma}\circ g}=I_{l,\bar{\sigma}}$ for all $g\in G_{K}$. Notice that $V_{\sigma}=V_{\sigma\circ g}$. Such $\mathcal{V}$ is called a pure realization of weight $n$.

In particular we have the identity realization and the Tate realization, respectively:

[TABLE]

In what follows, we will write $\mathcal{V}$, and sometimes $\mathcal{V}^{\prime}$, to denote an arbitrary element of $R_{K}$.

Remark 9.3*.*

See [Ja1, p. 11] for the definition of $\operatorname{Hom}_{R_{K}}(\mathcal{V},\,\mathcal{V}^{{}^{\prime}})$.

Definition 9.4**.**

The category $R_{K}^{\rm{p}}$ of polarized realizations is defined to be the full subcategory of $R_{K}$ generated, for all $n\in\mathbb{Z}$, by objects:

[TABLE]

where $(V_{\sigma},\psi_{\sigma})_{\sigma}$ are pure, polarized, rational Hodge structures of weight $n$, for any $n\in\mathbb{Z}$, and $(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$ is associated with $(V_{\sigma},\psi_{\sigma})_{\sigma}$ (cf. (3.2), (3.3) in §3) as follows:

[TABLE]

The family of $l$-adic representations $(V_{l},\psi_{l})_{l}$ is associated with $(V_{\sigma},\psi_{\sigma})_{\sigma}$ as follows:

[TABLE]

such that $g\circ I_{l,\bar{\sigma}\circ g}=I_{l,\bar{\sigma}}$ for all $g\in G_{K}$.

Lemma 9.5**.**

The category $R_{K}^{\rm{p}}$ is semisimple.

Proof.

Consider the subobject $(\mathcal{V}^{\prime},\psi^{\prime})\subset(\mathcal{V},\psi)$ i.e.:

[TABLE]

and $I_{\infty,\sigma}^{\prime}$ (resp. $I_{l,\bar{\sigma}}^{\prime}$) is the obvious restriction of ${I_{\infty,\sigma}}$ (resp. $I_{l,\bar{\sigma}}$). Because $\psi_{\sigma}$ is nondegenerate for each $\sigma$ $\psi_{{}_{\rm{DR}}}$ and $\psi_{l}$ for all $l$ are nondegenerate by (9.6) and (9.8). Let   $V_{{}_{\rm{DR}}}^{\prime\prime},\,\,V_{l}^{\prime\prime},\,\,V_{\sigma}^{\prime\prime}$ be the orthogonal complements of $V_{{}_{\rm{DR}}}^{\prime},\,\,V_{l}^{\prime},\,\,V_{\sigma}^{\prime}$ with respect to the forms $\psi_{{}_{\rm{DR}}},\,\,\psi_{l},\,\,\psi_{\sigma}$, respectively. Hence we obtain nondegenerate forms

[TABLE]

and a well-defined restriction $I_{\infty,\sigma}^{\prime\prime}$ (resp. $I_{l,\bar{\sigma}}^{\prime\prime}$) of ${I_{\infty,\sigma}}$ (resp. $I_{l,\bar{\sigma}}$). Hence:

[TABLE]

is a well-defined polarized realization and it is clear that

[TABLE]

Remark 9.6*.*

Actually we can prove more than Lemma 9.5. Namely, if $\mathcal{V}^{\prime}\in{\rm{obj}}(R_{K})$ and $\mathcal{V}^{\prime}\subset\mathcal{V}$ for $(\mathcal{V},\psi)\in{\rm{obj}}(R_{K}^{\rm{p}})$, then $\psi_{\sigma}^{\prime}:={\psi_{\sigma}}_{|V_{\sigma}^{\prime}}$ is a polarization on $V_{\sigma}^{\prime}$ because the Weil operator $C\colon V_{\sigma}\rightarrow V_{\sigma}$ restricts naturally to the Hodge structure $V_{\sigma}^{\prime}$. Now defining $\psi_{{}_{\rm{DR}}}^{\prime}={\psi_{{}_{\rm{DR}}}}_{|V_{{}_{\rm{DR}}}^{\prime}},\,\,\psi_{l}^{\prime}={\psi_{l}}_{|V_{l}^{\prime}}$ and observing that $I_{\infty,\sigma}$ (resp. $I_{l,\bar{\sigma}}$) naturally restricts to the comparison isomorphism ${I_{\infty,\sigma}}^{\prime}$ (resp. $I_{l,\bar{\sigma}}^{\prime}$) on $\mathcal{V}^{\prime}\subset\mathcal{V}$, we obtain the polarized realization $(\mathcal{V}^{\prime},\psi^{\prime})\subset(\mathcal{V},\psi)$.

The polarization $\psi\colon V\times V\rightarrow\mathbb{Q}(-n)$ leads to two isomorphisms $V(n)\,\,{\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}}\,\,V^{\vee}$ of $\mathbb{Q}$-vector spaces:

[TABLE]

This gives a $\mathbb{Q}$-bilinear, nondegenerate pairing:

[TABLE]

We define the action of Weil’s operator $C\colon V\rightarrow V$ on $V^{\vee}$ as follows. For $f\in V^{\vee}$ we put $C(f):=f\circ C$. Because $\psi$ is a polarization, the form

[TABLE]

is positive definite on $V^{\vee}\otimes_{\mathbb{Q}}\mathbb{R}$. Hence $\psi^{\vee}$ is a polarization. In the same way as above we construct natural nondegenerate pairings $\psi_{{}_{\rm{DR}}}^{\vee}$ and $\psi_{l}^{\vee}$. The comparison isomorphisms $I_{\infty,\sigma}^{\vee}$ and $I_{l,\bar{\sigma}}^{\vee}$ are the transposes of $I_{\infty,\sigma}$ and $I_{l,\bar{\sigma}}$ (see [Ja1, p. 15]). Hence the realization

[TABLE]

is in the category $R_{K}^{\rm{p}}$.

Remark 9.7*.*

Because $R_{K}^{\rm{p}}$ is a full subcategory of $R_{K}$, we will also write $\mathcal{V}$ instead of $(\mathcal{V},\psi)$ for objects of $R_{K}^{\rm{p}}$ whenever it does not lead to a conflict of notation or misunderstanding.

Proposition 9.8**.**

The category $R_{K}^{\rm{p}}$ is a $\mathbb{Q}$-rational, neutral, semisimple, Tannakian subcategory of $R_{K}$.

Proof.

By the above computations and [Ja1, pp. 12–15] we observe that if $\mathcal{V}$ and $\mathcal{W}$ are objects of $R_{K}^{\rm{p}}$ then $\mathcal{V}\otimes\mathcal{W}$ and $\underline{\operatorname{Hom}}(\mathcal{V},\mathcal{W})=\mathcal{V}^{\vee}\otimes\mathcal{W}$ are also objects of $R_{K}^{\rm{p}}$. Moreover it is clear that ${\mathcal{V}}^{\vee\vee}=\mathcal{V}$. Applying Lemma 9.5 finishes the proof. ∎

Fix $\sigma$ and put $V:=V_{\sigma}$. Consider the fiber functor $H:=H_{B}$:

[TABLE]

Now fix $\mathcal{V}\in{\rm{obj}}(R_{K}^{\rm{p}})$ and put $\mathcal{W}:=\mathcal{V}\oplus{\bf{1}}(1)\in{\rm{obj}}(R_{K}^{\rm{p}})$. Let $R_{K}[\mathcal{V}]$ (resp. $R_{K}[\mathcal{W}]$) denote the smallest strictly full Tannakian subcategory of $R_{K}^{\rm{p}}$ containing $\mathcal{V}$ (resp. containing $\mathcal{W}$). Consider the fiber functors $H_{\mathcal{V}}:=H_{|_{R_{K}[\mathcal{V}]}}$ and $H_{\mathcal{W}}:=H_{|_{R_{K}[\mathcal{W}]}}$. We put:

[TABLE]

Remark 9.9*.*

It follows directly from Definition 9.1 (see especially Diagram 9.1) applied to the Tannakian category $R_{K}[\mathcal{V}]$ (resp. $R_{K}[\mathcal{W}]$) that for every $\mathbb{Q}$-algebra $R$, each family $(\lambda_{\mathcal{X}})$, $\mathcal{X}\in{\rm{obj}}(R_{K}[\mathcal{V}])$ (resp. $\mathcal{X}\in{\rm{obj}}(R_{K}[\mathcal{W}]$)) is determined uniquely by $\lambda_{\mathcal{V}}$ and $\lambda_{\mathcal{V}^{\vee}}$ (resp. by $\lambda_{\mathcal{W}}$ and $\lambda_{\mathcal{W}^{\vee}}$).

Consider the $\mathbb{Q}_{l}$-linear, Tannakian, neutral category of finite dimensional, continuous, Galois representations ${\rm{Rep}}_{\mathbb{Q}_{l}}G_{K}$ with the forgetful fiber functor

[TABLE]

For the object $\mathcal{V}$ (resp. for the object $\mathcal{W}$) consider the $l$-adic realization $V_{l}$ (resp. the $l$-adic realization $W_{l}$). Let ${\rm{Rep}}_{\mathbb{Q}_{l}}G_{K}[V_{l}]$ (resp. ${\rm{Rep}}_{\mathbb{Q}_{l}}G_{K}[W_{l}]$) denote the smallest strictly full Tannakian subcategory of ${\rm{Rep}}_{\mathbb{Q}_{l}}G_{K}$ containing $V_{l}$ (resp. containing $W_{l}$). Let $H_{V_{l}}$ (resp. $H_{W_{l}}$) denote the fiber functor $H_{l}$ restricted to ${\rm{Rep}}_{\mathbb{Q}_{l}}G_{K}[V_{l}]$ (resp. ${\rm{Rep}}_{\mathbb{Q}_{l}}G_{K}[W_{l}]$).

Remark 9.10*.*

Again from Definition 9.1 applied to the Tannakian category ${\rm{Rep}}_{\mathbb{Q}_{l}}G_{K}[V_{l}]$ (resp. ${\rm{Rep}}_{\mathbb{Q}_{l}}G_{K}[W_{l}]$) it follows that for every $\mathbb{Q}_{l}$-algebra $R$ each family $(\lambda_{U_{l}})$, $U_{l}\in{\rm{obj}}({\rm{Rep}}_{\mathbb{Q}_{l}}G_{K}[V_{l}])$ (resp. $U_{l}\in{\rm{obj}}({\rm{Rep}}_{\mathbb{Q}_{l}}G_{K}[V_{l}])$ is determined uniquely by $\lambda_{V_{l}}$ and $\lambda_{V_{l}^{\vee}}$ (resp. by $\lambda_{W_{l}}$ and $\lambda_{W_{l}^{\vee}}$).

It is known [An1, section 7.1.3, p.70] that

[TABLE]

Lemma 9.11**.**

There are natural embeddings:

[TABLE]

Proof.

Consider $(\lambda_{U_{l}})\in{\rm{Aut}}^{\otimes}\,H_{V_{l}}(\mathbb{Q}_{l})$ (resp. $(\lambda_{U_{l}})\in{\rm{Aut}}^{\otimes}\,H_{W_{l}}(\mathbb{Q}_{l})$) with $U_{l}\in{\rm{obj}}({\rm{Rep}}_{\mathbb{Q}_{l}}G_{K}[V_{l}])$ (resp. $U_{l}\in{\rm{obj}}({\rm{Rep}}_{\mathbb{Q}_{l}}G_{K}[W_{l}])$). The main constraint on the family $(\lambda_{U_{l}})$ is commutativity of the following Diagram 9.2 for all $\alpha_{l}\in\operatorname{Hom}_{\mathbb{Q}_{l}[G_{K}]}(U_{l},U^{\prime}_{l})$:

In the family $(\lambda_{U_{l}})$ there is a subfamily $(\lambda_{X_{l}})$ where $X_{l}=H_{V_{l}}(\mathcal{X})$ with $\mathcal{X}\in{\rm{obj}}(R_{K}[\mathcal{V}])$ (resp. $\mathcal{X}\in{\rm{obj}}(R_{K}[\mathcal{W}]$) such that for all $X_{l}=H_{V_{l}}(\mathcal{X})$, $Y_{l}=H_{V_{l}}(\mathcal{Y})$, $\alpha_{l}=H_{V_{l}}(\alpha)$ with $\alpha\in\operatorname{Hom}_{(R_{K}[\mathcal{V}])}(\mathcal{X},\mathcal{Y})$ (resp. $X_{l}=H_{W_{l}}(\mathcal{X})$, $Y_{l}=H_{W_{l}}(\mathcal{Y})$, $\alpha\in\operatorname{Hom}_{(R_{K}[\mathcal{W}])}(\mathcal{X},\mathcal{Y})$) the following Diagram 9.3 commutes for all $l$:

Observe that $(I_{l,\bar{\sigma}})^{-1}(X_{l})=H_{\sigma}(\mathcal{X})\otimes_{\mathbb{Q}}\mathbb{Q}_{l}$ and $(I_{l,\bar{\sigma}})^{-1}(\alpha_{l})=H_{\sigma}(\alpha)\otimes 1$. Applying $(I_{l,\bar{\sigma}})^{-1}$ to the family $(\lambda_{X_{l}})$ and Diagram 9.3, we obtain a family $(\lambda_{\mathcal{X}})$ with $\lambda_{\mathcal{X}}:=(I_{l,\bar{\sigma}})^{-1}(\lambda_{X_{l}})$ and a commutative Diagram 9.4 for each $l$:

The association $(\lambda_{U_{l}})\,\mapsto\,(\lambda_{\mathcal{X}})$ described above gives natural embeddings:

[TABLE]

This finishes the proof because:

[TABLE]

The Tannakian categories $R_{K}[\mathcal{V}]$, $R_{K}[{\bf{1}}(1)]$, $R_{K}[\mathcal{W}]$ are semisimple as full subcategories of $R_{K}^{\rm{p}}$. It follows by [DM, Prop. 2.21 (a), Remark 2.29] that the following homomorphisms are epimorphisms:

[TABLE]

Consider the projections: $\pi_{\mathcal{V}}\colon\mathcal{W}\rightarrow\mathcal{V}$, $\pi_{{\bf{1}}(1)}\colon\mathcal{W}\rightarrow{\bf{1}}(1)$, $\pi_{\mathcal{V}^{\vee}}\colon\mathcal{W}^{\vee}\rightarrow\mathcal{V}^{\vee}$, $\pi_{{\bf{1}}(-1)}\colon\mathcal{W}^{\vee}\rightarrow{\bf{1}}(-1)$. Then ${\rm{Id_{\mathcal{W}}}}=\pi_{\mathcal{V}}+\pi_{{\bf{1}}(1)}$ and ${\rm{Id_{\mathcal{W}^{\vee}}}}=\pi_{\mathcal{V}^{\vee}}+\pi_{{\bf{1}}(-1)}$. Diagram 9.1 applied to the category $R_{K}[\mathcal{W}]$ and its subcategories $R_{K}[\mathcal{V}]$ and $R_{K}[{\bf{1}}(1)]$ gives:

[TABLE]

By Remark 9.9 the following homomorphism is a closed immersion:

[TABLE]

Consider $(\mathcal{V},\psi)\,=\,(\,(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}}),\,(V_{l},\psi_{l})_{l},\,(V_{\sigma},\psi_{\sigma})_{\sigma},\,I_{\infty,\sigma},\,I_{l,\bar{\sigma}})\in{\rm{obj}}(R_{K}^{\rm{p}})$. Define (cf. Definition 9.4 and [Ja1, Def. 2.16]):

[TABLE]

[TABLE]

where $(\overline{V}_{{}_{\rm{DR}}},\overline{\psi}_{{}_{\rm{DR}}}):=(V_{{}_{\rm{DR}}}\otimes_{K}\overline{K},\psi_{{}_{\rm{DR}}}\otimes_{K}\overline{K})$,   $(\overline{V}_{l},\overline{\psi}_{l})_{l}:=(V_{l},\psi_{l})_{l}$,   $\bar{I}_{l,\bar{\sigma}}:=I_{l,\bar{\sigma}}$   and   $(\overline{V}_{\bar{\sigma}},\overline{\psi}_{\bar{\sigma}})_{\bar{\sigma}}:=(V_{\sigma},\psi_{\sigma})_{\sigma}$,   $\bar{I}_{\infty,\bar{\sigma}}:=I_{\infty,\sigma}$,   $g\circ I_{l,\bar{\sigma}\circ g}=I_{l,\bar{\sigma}}$ for all ${l}$, for all $g\in G_{K}$, for all $\sigma\colon K\hookrightarrow\mathbb{C}$, and for all $\bar{\sigma}\colon\overline{K}\hookrightarrow\mathbb{C}$ such that $\bar{\sigma}_{|K}=\sigma$.

Definition 9.12**.**

In the following statements, put $D:=\operatorname{End}_{R_{\overline{K}}^{\rm{p}}}\,((\overline{\mathcal{V}},\overline{\psi}))$.

Proposition 9.13**.**

Let $K$ be a number field. Let

[TABLE]

be a polarized, pure realization with components satisfying conditions (D1), (D2), (DR1), (DR2), (R1)–(R4) of §2, §3, and §4 respectively with $D$ as in Definition 9.12. Then Conjectures 8.1(a) and 8.2(a) hold for the family $(V_{l})_{l}$.

Proof.

This follows by Lemma 9.11 and properties of morphisms (9.9), (9.10), (9.11). ∎

Corollary 9.14**.**

Let $K$ be a number field. Let

[TABLE]

be a polarized, pure realization with components satisfying conditions (D1), (D2), (DR1), (DR2), (R1)–(R4) of §2, §3, and §4 respectively with $D$ as in Definition 9.12. Then Conjectures 8.1, 8.2, 7.5, and 7.19 are all equivalent for the family $(V_{l})_{l}$.

Proof.

It follows from Proposition 9.13 and Corollary 8.18. ∎

10. Remarks on equidistribution

This section is logically independent of the preceding ones. In particular, we do not need any of the assumptions (D1), (D2), (DR1), (DR2), (R1)–(R4). We start with some general facts about pushforward measures in measure theory.

Remark 10.1*.*

Let $\phi\colon(X,\Sigma)\rightarrow(X^{\prime},\Sigma^{\prime})$ be a measurable map with $\sigma$-algebras $\Sigma$ and $\Sigma^{\prime}$ respectively. If $\mu$ is a measure on $X$, then the pushforward measure $\phi_{\ast}\mu$ on $X^{\prime}$ is defined as follows:

[TABLE]

Hence for every measurable function $f\colon X^{\prime}\rightarrow\mathbb{C}$ we obtain:

[TABLE]

If $\mu$ is a probabilistic measure, then the pushforward measure $\phi_{\ast}\mu$ is also probabilistic. If $\phi\colon X\rightarrow X^{\prime}$ is a continuous map between topological spaces $X$ and $X^{\prime}$ and the $\sigma$-algebras $\Sigma$, $\Sigma^{\prime}$ are Borel, then $\phi$ is a measurable map. Important special cases of this are discussed in the following Remark 10.2.

Remark 10.2*.*

Let $\pi\colon X\rightarrow X^{\prime}$ be a surjective map of sets. If $X$ is a topological space (resp. $(X,\Sigma)$ is a measurable space with $\sigma$-algebra $\Sigma$), we can naturally induce on $X^{\prime}$ a quotient topology so that $\pi$ becomes continuous (resp. we can naturally induce a $\sigma$-algebra $\Sigma^{\prime}$ on $X^{\prime}$ so that $\pi\colon(X,\Sigma)\rightarrow(X^{\prime},\Sigma^{\prime})$ becomes a measurable map) because $\pi^{-1}$ commutes with complement and arbitrary unions and intersections of sets. In this case, $f\colon X^{\prime}\rightarrow\mathbb{C}$ is continuous (resp. measurable) iff $f\circ\pi$ is continuous (resp. measurable).

In particular, if $G$ is a compact group and $U\triangleleft G$ is a normal open subgroup, thne the quotient $G/U$ is a finite group and the induced topology on $G/U$ via the quotient map $\pi\colon G\rightarrow G/U$ is the discrete topology. If $\Sigma_{G}$ is the Borel $\sigma$-algebra on $G$ and $\mu$ is the Haar measure on $G$, then the induced Borel $\sigma$-algebra $\Sigma_{G/U}$ on $G/U$ is $2^{G/U}$ and the pushforward measure $\pi_{\ast}\mu$ on $G/U$ is the atomic (or discrete) measure.

Remark 10.3*.*

For any epimorphism $\pi\colon G\rightarrow G^{\prime}$ of compact Lie groups, we have $\pi_{\ast}\mu=\mu^{\prime}$ where $\mu$ (resp. $\mu^{\prime}$) is the probabilistic Haar measure on $G$ (resp. on $G^{{}^{\prime}}$) [Hal, Theorem C, Chap. XII].

Remark 10.4*.*

Let $G$ be a compact group with Haar measure $\mu$ and let $X(G)$ be the set of conjugacy classes of $G$. Consider the natural map $\pi\colon G\rightarrow X(G)$. If $f\colon G\rightarrow\mathbb{C}$ is a class function, we will also denote by $f$ the naturally induced function $X(G)\rightarrow\mathbb{C}$. By (10.1), for a measurable class function $f\colon G\rightarrow\mathbb{C}$ we have:

[TABLE]

Let $G$ be a compact group with probabilistic (i.e., normalized) Haar measure $\mu$. Let $G_{0}\triangleleft G$ be an open subgroup. Let $\mu_{0}$ denote the probabilistic Haar measure of $G_{0}$, so that $\mu_{0}=[G:\,G_{0}]\,\mu_{|G_{0}}$. The measure induced by $\mu$ on $X(G)$ (resp. $\mu_{0}$ on $X(G_{0})$) will also be denoted $\mu$ (resp. $\mu_{0}$) cf. Remark 10.4. The group $G$ can be decomposed into a finite number of cosets with respect to $G_{0}$:

[TABLE]

Remark 10.5*.*

Let $(x_{n})$ be a sequence of elements of $X(G)$. By definition, the sequence $(x_{n})$ is equidistributed in $X(G)$ if and only if for each continuous class function $f$ of $G$,

[TABLE]

By the Peter-Weyl theorem (see [Se1, Appendix to Chapter I]), we may further say that the sequence $(x_{n})$ is equidistributed in $X(G)$ if and only if for each nontrivial irreducible character $\chi$ of $G$,

[TABLE]

Let $\rho\colon G\rightarrow GL(V)$ be the irreducible representation with character $\chi$, where $V$ is a complex vector space of dimension $d$. Because $G$ is compact, for each $g\in G$, $\chi(g)$ is a sum of $d$ complex numbers of absolute value 1, and hence $|\chi(g)|\leq d$.

Consider the inclusion homomorphism $j\colon G_{0}\to G$. The map induced by $j$ on the sets of conjugacy classes of $G_{0}$ and $G$ will be denoted as follows:

[TABLE]

Remark 10.6*.*

The map $X(j)$ is not necessarily an inclusion (that is, there may be fusion of conjugacy classes from $G_{0}$ to $G$), but is finite-to-one. In any case, given a sequence of elements of $X(G)$ belonging to the image of $X(G_{0})$, we may still ask whether this sequence is equidistributed for the image (pushforward) of $\mu_{0}$.

The following proposition is of independent interest, but its proof presents a strategy we will apply later in this section.

Proposition 10.7**.**

Let $(x_{i})$ be a sequence of elements of $X(G)$ that is equidistributed with respect to $\mu$. Put $k_{n}:=\sharp\{x_{i}\in\mathrm{image}(X(j))\colon 1\leq i\leq n\}$. Assume that:

[TABLE]

Then the subsequence $(x_{n_{k}})$ of elements of $(x_{i})$ that are contained in the image of $X(j)$ is equidistributed with respect to the pushforward of $\mu_{0}$.

Proof.

We equate equidistribution in $X(G)$ with a statement about continuous class functions on $G$ as per Remark 10.5; similarly, we equate equidistribution in the image of $X(j)$ with a statement about continuous class functions on $G_{0}$ which are constant on fibers of the map $X(j)$. Let $f\colon G\rightarrow\mathbb{C}$ be any continuous class function. Observe that $f_{0}:=f\circ j\colon G_{0}\rightarrow\mathbb{C}$ is a continuous class function on $G_{0}$. Let $f_{1}\colon G\rightarrow\mathbb{C}$ be the extension of $f_{0}$ by [math] beyond $G_{0}$. The continuous function $f_{1}$ is a class function on $G$ because $G_{0}\triangleleft G$. By assumption,

[TABLE]

By the construction of $f_{1}$, the equation (10.6) assumes the following form:

[TABLE]

Hence:

[TABLE]

Since

[TABLE]

and the limits of the numerator and denominator on the right exist by assumptions (see (10.5) and (10.6)), the limit on the left also exists and by (10.1), (10.5), and (10.8) we get:

[TABLE]

as desired. ∎

For the rest of the section, we study the problem of inverting Proposition 10.7, which is to say deducing an equidistribution statement in $X(G)$ from equidistribution statements in $X(G_{0})$ and $X(G/G_{0})$. Our main tool for this will be induced characters.

Lemma 10.8**.**

Let $\chi_{0}$ be a character of $G_{0}$ and set $\chi:=\operatorname{Ind}_{G_{0}}^{G}\,(\chi_{0})$. Then

[TABLE]

Proof.

By [Se3, Theorem 12] and [Se3, p. 34]:

[TABLE]

Observe that $g_{i}^{-1}gg_{i}\in G_{0}$ if and only if $g\in g_{i}G_{0}g_{i}^{-1}=G_{0}$ because $G_{0}\triangleleft G$. In the following computations, we treat $\chi_{0}$ as a function on $G$ by extending it by [math] outside $G_{0}$. With this convention in mind, by applying (10.11) we obtain:

[TABLE]

Making the substitutions $g\mapsto g_{i}gg_{i}^{-1}$ for each $i$ under the integrals on the right of (10.12), then applying the right and left translation invariance of $\mu$, we obtain:

[TABLE]

because the function $g\mapsto\chi_{0}(g_{i}gg_{i}^{-1})$ has support in $G_{0}$ and $\mu_{0}=[G:\,G_{0}]\,\mu_{|G_{0}}$. ∎

Lemma 10.9**.**

Let $\chi_{0}$ be a character of $G_{0}$ and $\chi:=\operatorname{Ind}_{G_{0}}^{G}\,(\chi_{0})$. Then

[TABLE]

where $j\colon G_{0}\rightarrow G$ is the inclusion homomorphism.

Proof.

By (10.1) we obtain:

[TABLE]

[TABLE]

Again treating $\chi_{0}$ as a function on $G$ by extending it by [math] outside $G_{0}$ and then applying the right and left translation invariance of $\mu$ we get: (cf. the proof of Lemma 10.8)

[TABLE]

[TABLE]

∎

Remark 10.10*.*

We will use Lemmas 10.8 and 10.9 in the following fashion. In (10.3), suppose that $\chi$ is the character associated to the representation $\rho$; assume also that (10.5) holds. If $\rho$ is induced from a representation $\rho_{0}$ of $G_{0}$, we may equate the desired equality with the corresponding equality for the character of $\rho_{0}$, using (10.5) to compare the left-hand sides and Lemmas 10.8 and 10.9 to compare the right-hand sides.

Lemma 10.11**.**

Let $(x_{n})$ be a sequence of elements of $X(G)$. Suppose that:

1. (1)

The image of the sequence $(x_{n})$ in $X(G/[G,G])$ is equidistributed for the probabilistic Haar measure on $G/[G,G]$. 2. (2)

Every irreducible character $\chi$ of $G$ of dimension $>1$ is induced from an irreducible character $\chi_{0}$ of $G_{0}$. 3. (3)

The equation (10.5) holds. 4. (4)

The subsequence of elements of $(x_{n})$ that are contained in the image of $X(j)$ is equidistributed with respect to the pushforward of $\mu_{0}$.

Then $(x_{n})$ is equidistributed in $X(G)$ with respect to $\mu$.

Proof.

For $\chi$ of dimension 1, $\chi$ is abelian and thus factors through the epimorphism $G\rightarrow G/[G,G]$. Hence this case is covered by condition (1) and Remark 10.3.

For $\chi$ of dimension $>1$, let $(x_{n_{k}})$ be the subsequence of elements of $(x_{n})$ that are contained in the image of $X(j)$. By assumption (4) we have:

[TABLE]

By (2), (3), (4) and Lemmas 10.8 and 10.9, the equality (10.14) is equivalent to:

[TABLE]

because $\chi(x_{i})=0$ for $x_{i}$ not in the subsequence $(x_{n_{k}})$. ∎

Proposition 10.12**.**

Let $(x_{n})$ be a sequence of elements of $X(G)$. Let $(x_{n_{k}})$ be the subsequence of elements of $(x_{n})$ in the image of $X(j)$.

• (a)

Each fiber of $x_{n_{k}}$ of the map $X(j)$ equals $\{g_{i}y_{n_{k}}g_{i}^{-1}\colon 0\leq i\leq t\}$, where $y_{n_{k}}\in X(G_{0})$ is an element of this fiber.

• (b)

If assumptions (1)–(3) of Lemma 10.11 hold and the sequence $(g_{i}y_{n_{k}}g_{i}^{-1})_{i,k}$ is equidistributed in $X(G_{0})$ with respect to $\mu_{0}$, then $(x_{n})$ is equidistributed in $X(G)$ with respect to $\mu$.

Proof.

The claim (a) is obvious. To prove (b), assumption (1) of Lemma 10.11 again handles the case where $\chi$ is a character of dimension 1, so we may assume that $\chi$ is of dimension $>1$. The number of elements in the sequence $(g_{i}y_{n_{k}}g_{i}^{-1})_{i,j}$, for $1\leq k\leq k_{n}$ and $0\leq i\leq t$, asymptotically equals $[G:\,G_{0}]\,k_{n}$ by assumption (3) of Lemma 10.11. Hence by (10.11) and equidistribution of $(g_{i}y_{n_{k}}g_{i}^{-1})_{i,j}$ we have:

[TABLE]

Multiplying (10.16) by $[G:\,G_{0}]$ and applying (10.11) and Lemma 10.9, we observe that (10.16) is equivalent to (10.14). Hence the Proposition follows by Lemma 10.11. ∎

Remark 10.13*.*

A basic example to which Lemma 10.11 applies is where $G_{0}=\operatorname{SO}(2)$ and $G$ is the normalizer of $\operatorname{SO}(2)$ in $\operatorname{SU}(2)$.

When Lemma 10.11 does not apply, we can still make a nontrivial reduction, loosely inspired by [Jo].

Because $G$ and $G_{0}$ are compact, $G_{0}$ is open in $G$, and $G/G_{0}$ is finite, then each subgroup $H$ such that $G_{0}\subset H\subset G$ is automatically compact and open in $G$. Let $\mu_{H}$ be the probabilistic Haar measure of $H$, which satisfies $\mu_{H}=[G:\,H]\,\mu_{|H}$. We can generalize Lemma 10.8 as follows.

Lemma 10.14**.**

Let $\chi$ be a character of $H$. Then

[TABLE]

Proof.

By [Se3, Theorem 12] and [Se3, p. 34]:

[TABLE]

Observe that $g_{i}^{-1}gg_{i}\in H$ if and only if $g\in g_{i}Hg_{i}^{-1}$. In the following computations, we may treat the function $g\mapsto\chi(g_{i}^{-1}gg_{i})$ as a function on $G$ with support in $g_{i}Hg_{i}^{-1}$. Hence by applying (10.18), making the substitution $g\mapsto g_{i}gg_{i}^{-1}$ for each $i$, and applying right and left shift invariance of $\mu$, we obtain:

[TABLE]

Definition 10.15**.**

Let $H$ be a subgroup of $G$ such that $G_{0}\subset H\subset G$. Let $\lambda$ be a character of $H/G_{0}$. Let $\widetilde{\lambda}$ denote its lift to the character of $H$, so that for each $h\in H$:

[TABLE]

Lemma 10.16**.**

Let $H$ be a subgroup of $G$ such that $G_{0}\subset H\subset G$. Then

[TABLE]

Proof.

Observe that $G=\bigsqcup_{i=1}^{m}g_{i}H$ if and only if $G/G_{0}=\bigsqcup_{i=1}^{m}g_{i}G_{0}H/G_{0}$. Hence:

[TABLE]

Remark 10.17*.*

Let $H\subset G$ and let $\chi$ (resp. $\lambda$) be a character of $G$ (resp. $H$). Then there is the following formula:

[TABLE]

Lemma 10.18**.**

Every character of $G$ is a $\mathbb{Q}$-linear combination of characters of the form $\operatorname{Ind}^{G}_{G_{1}}\chi_{1}$ where $G_{1}$ is the preimage in $G$ of a cyclic subgroup of $G/G_{0}$ and $\chi_{1}$ is an irreducible character of $G_{1}$, not induced from any proper subgroup of $G_{1}$ containing $G_{0}$, whose restriction to $G_{0}$ is irreducible.

Proof.

Let $1_{G/G_{0}}$ denote the trivial character of $G/G_{0}$. Naturally $\widetilde{1_{G/G_{0}}}=1_{G}$. By Artin’s theorem on induced characters [Se3, Chapter IX, Theorem 17], there exist $a_{i}\in\mathbb{Q}$, cyclic subgroups $H_{i}/G_{0}$ of $G/G_{0}$, and characters $\lambda_{i}$ of $H_{i}/G_{0}$ such that

[TABLE]

Lifting the equation (10.22) to $G$ and applying Lemma 10.16, we obtain:

[TABLE]

By (10.21), for any character $\chi$ of $G$, we obtain:

[TABLE]

We can write $\widetilde{\lambda_{i}}\cdot{{\rm{Re}}}_{H_{i}}^{G}\chi=\sum_{j=1}^{t_{i}}\chi_{i,j}$where $\chi_{i,j}$ are irreducible characters of $H_{i}$. Hence by (10.24) we obtain:

[TABLE]

where $\chi_{i,j}$ are irreducible. We can also assume that $\chi_{i,j}$ is not induced from any proper subgroup of $H_{i}$ containing $G_{0}$ because induction is transitive. This presents $\chi$ as a $\mathbb{Q}$-linear combination of characters of the form $\operatorname{Ind}^{G}_{G_{1}}\chi_{1}$ where $G_{1}$ is the preimage in $G$ of a cyclic subgroup of $G/G_{0}$ and $\chi_{1}$ is irreducible, not induced from any proper subgroup of $G_{1}$ containing $G_{0}$.

To complete the argument, we may assume that $G/G_{0}$ is itself cyclic and that $\chi$ is an irreducible character not induced from any proper subgroup of $G$ containing $G_{0}$. Let $\rho$ be the associated representation. By [Se3, Prop. 24], $\rho|_{G_{0}}$ is isotypic; let $\tau$ be an irreducible subrepresentation of $\rho|_{G_{0}}$. By the second theorem of Clifford [C-R, Theorem 11.20] cf. [Ko, p. 493], the associated projective representation $\overline{\rho}$ of $\rho$ factors as a tensor product $\overline{\rho}_{1}\otimes\overline{\rho}_{2}$ in which $\overline{\rho}_{1}|_{G_{0}}$ is isomorphic to $\overline{\tau}$. Moreover, $\ker(\overline{\rho}_{1})$ is the centralizer of $G_{0}/(G_{0}\cap\ker(\rho))$ in $G$ while $\ker(\overline{\rho}_{2})$ contains $G_{0}$. Since $\rho$ is irreducible and $G/G_{0}$ is cyclic, $\overline{\rho}_{2}$ must be the trivial representation, and so $\rho|_{G_{0}}$ is irreducible. ∎

Lemma 10.19**.**

Let $(x_{n})$ be a sequence of elements of $X(G)$. Suppose that for every subgroup $G_{1}$ of $G$ containing $G_{0}$ such that $G_{1}/G_{0}$ is cyclic, the subsequence of elements of $(x_{n})$ that are contained in the image of $X(G_{1})\to X(G)$ is equidistributed with respect to the pushforward of the probabilistic Haar measure of $G_{1}$. Then $(x_{n})$ is equidistributed in $X(G)$ with respect to $\mu$.

Proof.

This follows immediately from Remark 10.10 and Lemma 10.18. ∎

In the remainder of this section, we explore the obstruction to deducing equidistribution on $G$ from equidistribution on $G_{0}$.

Theorem 10.20**.**

Assume that $G_{0}$ is a connected Lie group, the simple factors of the derived group of $G_{0}$ are pairwise distinct, and none of these factors admits a nontrivial diagram automorphism. Let $(x_{n})$ be a sequence of elements of $X(G)$. Suppose that for every subgroup $G_{1}$ of $G$ containing $G_{0}$ such that $G_{1}/G_{0}$ is cyclic and acts trivially on $G_{0}$, the subsequence of elements of $(x_{n})$ that are contained in the image of $X(G_{1})\to X(G)$ is equidistributed with respect to the pushforward of the probabilistic Haar measure of $G_{1}$. Then $(x_{n})$ is equidistributed in $X(G)$ with respect to $\mu$.

Proof.

We check (10.3) for a particular irreducible representation $\rho$ with character $\chi$. By Lemma 10.18, we may assume that $G/G_{0}$ is cyclic, $\rho$ is injective, and $\rho|_{G_{0}}$ is irreducible. Let $Z$ be the center of $G_{0}$ and let $H$ be the derived group of $G_{0}$; then $H$ is semisimple and $HZ=G_{0}$ (see [Sep, Theorem 5.22]). Consequently, $\rho|_{H}$ is irreducible and $H$ is characteristic in $G$; in particular, we have a restriction homomorphism $\operatorname{Aut}(G_{0})\to\operatorname{Aut}(H)$ which induces a homomorphism $\operatorname{Out}(G_{0})\to\operatorname{Out}(H)$.

Using highest weight vectors, we obtain a canonical bijection between the isomorphism classes of irreducible representations of $H$ and the quotient of the weight lattice $L$ of $H$ by the action of the Weyl group $W$. The outer automorphism group $\operatorname{Out}(H)$ acts faithfully on $L/W$. The conjugation action $G\to\operatorname{Aut}(H)$ induces a homomorphism $G/G_{0}\to\operatorname{Out}(H)$ and hence an action of $G/G_{0}$ on $L/W$; this action must fix the vector corresponding to $\rho|_{H}$. Moreover, the images of $G/G_{0}$ in $\operatorname{Out}(G_{0})$ and $\operatorname{Out}(H)$ are isomorphic because $Z$ is central in $G$.

Now our condition that $H$ has pairwise distinct simple factors, none of which admits a nontrivial diagram automorphism, implies that $\operatorname{Out}(H)$ is trivial. By the previous paragraph, this means that $G/G_{0}$ is generated by an element that centralizes $G_{0}$. Hence this case is covered by our input hypothesis. ∎

Remark 10.21*.*

In Theorem 10.20, the restrictions on the derived group cannot be lifted. We give two illustrative examples of representations that cannot be handled when the restrictions are lifted.

• •

Take $H:=\operatorname{SU}(3)$, $G_{0}=ZH$ for $Z$ either a finite cyclic group or $U(1)$, $G:=G_{0}\rtimes\mathbb{Z}/2\mathbb{Z}$ for the nontrivial action on $H$ (fixing $Z$), and $\rho$ to be the adjoint representation. The induced homomorphism $G/G_{0}\to\operatorname{Out}(G_{0})$ is nontrivial, and the highest weight vector of $\rho$ is fixed by the action of $G/G_{0}$.

• •

Take $H=G_{0}:=\operatorname{SU}(2)\times\operatorname{SU}(2)$, $G:=\operatorname{Aut}(G_{0})\simeq G_{0}\rtimes\mathbb{Z}/2\mathbb{Z}$, and $\rho$ to be the external product of the standard representations of the two factors. This type of example can be regarded as a tensor induction as in [Ko]; however, this does not lead to a reduction for the equidistribution problem.

11. Application to the Sato–Tate conjecture

We next specialize the previous discussion to the setting of Sato–Tate groups. First, we make some general remarks about the relationship between equidistribution and $L$-functions, again following [Se1, Appendix to Chapter I].

Definition 11.1**.**

Let $\rho\colon G\to\operatorname{GL}(V)$ be a continuous representation of a compact topological group on a finite-dimensional $\mathbb{C}$-vector space $V$. Let $L/K$ be a finite extension of number fields and let $\Pi:=\Pi_{L}$ denote the set of all prime ideals in the ring of integers $\mathcal{O}_{L}$. Let $(x_{v})$ be a sequence of elements of $X(G)$ indexed by the prime ideals $v$ of $\mathcal{O}_{L}$. Define the $L$-function:

[TABLE]

This product is absolutely convergent for $\mathrm{Re}(s)>1$, and so defines a nowhere-vanishing holomorphic function in this region. For any subset $T$ of $\Pi$, we also set:

[TABLE]

Lemma 11.2**.**

Let $S$ denote any set of ideals in $\Pi$ whose symmetric difference with the set of prime ideals of $L$ of degree $1$ over $K$ is finite. Then $L_{\Pi-S}(s,\,\rho)$ defines a nowhere-vanishing analytic function on the region ${\rm{Re}}(s)>\frac{1}{2}$.

Proof.

For every $v\in\Pi$ we have:

[TABLE]

where the $\xi_{v,j}$ are eigenvalues of $\rho(x_{v})$ and are complex numbers of absolute value 1. For almost all $v\in\Pi-S$, we have $Nv=p_{v}^{r_{v}}$ with $r_{v}\geq 2$ where $v\,|\,p_{v}$ and $p_{v}$ is a prime number. Hence:

[TABLE]

This infinite product is absolutely convergent for ${\rm{Re}}(s)>\frac{1}{2}$ because the series

[TABLE]

is absolutely convergent for ${\rm{Re}}(s)>\frac{1}{2}$. ∎

In the notation of Lemma 11.2 we obtain:

[TABLE]

Lemma 11.3**.**

For $S$ as in Lemma 11.2, the sequence $(x_{v})$ is equidistributed with respect to the measure $\mu$ on $X(G)$ if and only if the sequence $(x_{v})_{S}$ is equidistributed with respect to the measure $\mu$ on $X(G)$.

Proof.

By (10.5), the sequence $(x_{v})$ is equidistributed with respect to the measure $\mu$ on $X(G)$ if and only if for each nontrivial irreducible representation $\rho$,

[TABLE]

Similarly, the sequence $(x_{v})_{S}$ is equidistributed with respect to the measure $\mu$ on $X(G)$ if and only if for each $\rho$,

[TABLE]

To equate these, note that by Lemma 11.2 applied to the trivial representation (evaluating at $s=1$), the sum

[TABLE]

is absolutely convergent. This implies firstly that

[TABLE]

and secondly that for any $\rho$,

[TABLE]

This yields the desired equivalence. ∎

Lemma 11.4**.**

Let $\mathcal{G}$ be an algebraic group over a field of characteristic [math]. Then:

• (a)

Every unipotent subgroup $U\subset\mathcal{G}$ is connected.

• (b)

For every unipotent element $u\in\mathcal{G}$ there is a smallest unipotent subgroup of $\mathcal{G}$ containing $u$.

• (c)

The set of unipotent elements $U(\mathcal{G})\subset\mathcal{G}$ is Zariski closed in $\mathcal{G}$.

• (d)

$U(\mathcal{G})\subset\mathcal{G}^{0}$.

Proof.

(a), (b), and (c) are well known. Containment (d) follows from (a) and (b). ∎

For the rest of this section we assume Conjectures 7.5 and 7.19. Therefore we can work under the assumptions of Theorem 7.17 and Theorem 7.26; consequently, we return to all of the assumptions of sections 2–8. Let

[TABLE]

Let

[TABLE]

By (7.11) and (7.20) and Propositions 7.9 and 7.21, applied for the base fields $K$ and $K_{0}$, we obtain the following natural isomorphisms of groups:

[TABLE]

Hence we obtain the following equality:

[TABLE]

Observe that (cf. Corollary 7.31):

[TABLE]

By (7.13) or (7.22), $G_{0}$ is open in $G$. We may thus set notation as in §10; in particular, let $\mu$ be the probabilistic Haar measure on $G$ and $\mu_{0}:=[K_{0}:\,K]\,\mu_{|G_{0}}$ be the corresponding probabilistic Haar measure on $G_{0}$.

Let $S_{\mathrm{fin}}$ be the set of primes $v\subset\mathcal{O}_{K}$ that are ramified in the representation $\rho_{l}$ of $G_{K}$ (4.1) together with the primes over $l$. For the prime $v\notin S_{\mathrm{fin}}$ (resp. $w\subset\mathcal{O}_{K_{0}}$ not over a prime in $S_{\mathrm{fin}}$), choose a prime $\overline{v}\subset\mathcal{O}_{\overline{K}}$ dividing $v$ (resp. $\overline{w}\subset\mathcal{O}_{\overline{K}}$ dividing $w$). Let $f_{v}\in\mathcal{G}$ (resp. $f_{w}\in\mathcal{G}_{0}$) denote the normalized Frobenius element for $\overline{v}\,|\,v$ (resp. the normalized Frobenius element for $\overline{w}\,|\,w$), see Remarks 7.11 and 7.32. We have $f_{v}=s_{v}u_{v}$ where $s_{v}\in\mathcal{G}$ (resp. $u_{v}\in\mathcal{G}$) is the semisimple part of $f_{v}$ (resp. is the unipotent part of $f_{v}$). In the same way, we obtain a factorization $f_{w}=s_{w}u_{w}$ with $s_{w}\in\mathcal{G}_{0}$ semisimple and $u_{w}\in\mathcal{G}_{0}$ unipotent. By Lemma 11.4, the containment (11.4), and the equality (11.3), we observe that:

[TABLE]

Proposition 11.5**.**

The image of the sequence $(f_{v})$ in $X(G/G_{0})$ is equidistributed for the probabilistic Haar measure on $G/G_{0}$.

Proof.

Via the isomorphism $\operatorname{Gal}(K_{0}/K)\simeq G/G_{0}$ (11.2), the claim translates into the Chebotarev density theorem. ∎

By equalities (7.12) and (7.13) of Theorem 7.17 or by equalities (7.21) and (7.22) of Theorem 7.26 we obtain:

[TABLE]

with $g_{0}=e$ for fixed coset representatives $g_{0},\dots,g_{t}$. These representatives may be taken to be images of elements of the Galois group $G_{K}$ via the $l$-adic representation. This choice of ${g_{i}}$’s is possible by loc. cit.

Let $\chi_{0}$ be a character of $G_{0}$ and let $\chi:={{\rm{Ind}}_{G_{0}}^{G}\,\chi_{0}}$. Recall the formula (10.11):

[TABLE]

Because $G_{0}\triangleleft G$ we obtain:

[TABLE]

It is clear due to (11.5) that for each $0\leq i\leq t$:

[TABLE]

Hence by (11.7) the elements of the subsequence $({f_{v_{i}}})_{v_{i}}$ of the sequence $(f_{v})$ such that $f_{v_{i}}\in g_{i}\,\mathcal{G}_{0}$ are in one-to-one correspondence with the elements of the subsequence $({s_{v_{i}}})_{v_{i}}$ of the sequence $(s_{v})$ such that $s_{v_{i}}\in g_{i}\,G_{0}$ for each $i=0,\dots,t$. By (10.11), (11.7), and (11.6) we obtain:

[TABLE]

On the other hand

[TABLE]

where $s_{w_{0,j}}=g_{j}^{-1}\,s_{v_{0}}\,g_{j}\in G_{0}$ for each prime ideal $w_{0,j}$ over $v_{0}$, by the choice of ${g_{j}}$’s. The primes $v_{0}$ are the only primes in the sequence $(v)$ outside $S_{\mathrm{fin}}$ that split completely in $\mathcal{O}_{K_{0}}$ because $s_{v_{0}}\in G_{0}$ due to (11.5).

Going towards the equidistribution questions, let $m\in\mathbb{N}$ and consider the sum:

[TABLE]

Put $N_{K}\,(m):=\,\#\{v_{i}\colon Nv_{i}\,\leq\,m,\,0\leq i\leq t\}$. Observe that $N_{K}\,(m)$ is asymptotically equal to

[TABLE]

and this is equal to

[TABLE]

By Lemma 10.8 and (11.9) the following equalities (11.10) and (11.11) are equivalent:

[TABLE]

Let $(s_{v})$ (resp. $(s_{w}))$ denote the sequence in $X(G)$ of conjugacy classes of elements $s_{v}$ (resp. the sequence in $X(G_{0})$ of conjugacy classes of elements $s_{w}$) by a slight abuse of notation.

Lemma 11.6**.**

If $(s_{v})$ is equidistributed in $X(G)$ with respect to $\mu$, then $(s_{w})$ is equidistributed in $X(G_{0})$ with respect to $\mu_{0}$.

Proof.

Let $(s_{v_{0}})$ be the subsequence of $(s_{v})$ with $s_{v_{0}}\in X(G_{0})$. The primes $v_{0}$ are precisely those that split completely in $K_{0}/K$. The sequence $(s_{w_{0,j}})_{0,j}$ is obtained from $(s_{v_{0}})$ by replacing each $s_{v_{0}}$ with $t+1$ conjugates $s_{w_{0,j}}=g_{j}^{-1}\,s_{v_{0}}\,g_{j}$. Let $\chi_{0}$ be an irreducible character of $G_{0}$ and $\chi:={{\rm{Ind}}_{G_{0}}^{G}\,\chi_{0}}$. By assumption and the definition of equidistribution, (11.10) holds. Hence the sequence $(s_{w_{0,j}})_{0,j}$ is equidistributed in $X(G_{0})$ with respect to $\mu_{0}$ because of (11.11). By Lemma 11.3, the sequence $(s_{w})_{w}$ is equidistributed in $X(G_{0})$ with respect to $\mu_{0}$. ∎

Theorem 11.7**.**

Assume that every nontrivial, irreducible character of $G$ with nonabelian image is induced from an irreducible character of $G_{0}$. Then the sequence $(s_{v})$ is equidistributed in $X(G)$ with respect to $\mu$ if and only if the sequence $(s_{w})$ is equidistributed in $X(G_{0})$ with respect to $\mu_{0}$.

Proof.

Equidistribution of $(s_{v})$ implies equidistribution of $(s_{w})$ by Lemma 11.6. We deduce the converse implication applying Lemma 10.11: condition (i) follows from the analytic continuation of abelian $L$-functions associated to Hecke characters (see [Jo, Corollary 15]); conditions (ii) and (iii) hold by hypothesis; condition (iv) holds by the Chebotarev density theorem. ∎

Our next goal is the strengthening of Theorem 11.7. Let $H$ be a subgroup of $G$, not necessarily normal, such that $G_{0}\subset H\subset G$. Via isomorphism $\operatorname{Gal}(K_{0}/K)\simeq G/G_{0}$ we have $\operatorname{Gal}(K_{0}/L)\simeq H/G_{0}$ for some subfield $L$ of $K_{0}$ containing $K$. Let $f_{v}\in\mathcal{G}$, as before, be the normalized Frobenius for $\overline{v}\,|\,v$ and let $f_{w_{1}}\in\mathcal{H}$ be the normalized Frobenius for $\overline{w_{1}}\,|\,w_{1}$ for a prime $w_{1}\subset\mathcal{O}_{L}$, outside the finite set of primes over $S_{\mathrm{fin}}$. Recall that by (7.11), we have a natural isomorphism

[TABLE]

Similarly to (10.2) we can write

[TABLE]

for fixed left coset representatives $g_{0},\dots,g_{t}\in G$ with $g_{0}=e$. Again these representatives may be taken to be images of elements of the Galois group $G_{K}$ via the $l$-adic representation. This choice of ${g_{i}}$’s is possible by equality (7.15) of Corollary 7.18 or equality (7.25) of Corollary 7.27.

Let:

[TABLE]

By Propositions 7.9 and 7.21, applied for base fields $K$ and $L$ we obtain the following, natural bijection of coset spaces:

[TABLE]

Hence we have:

[TABLE]

Therefore

[TABLE]

for the left coset representatives $g_{0},\dots,g_{t}$ with $g_{0}=e$ (see (11.13)).

Lemma 11.8**.**

There is a one-to-one correspondence between the set of elements $g_{i}\in\mathcal{G}$ such that $g_{i}^{-1}f_{v}g_{i}\in\mathcal{H}$ and the set of primes $w\,|\,v$ of $\mathcal{O}_{L}$ that split completely over $v$, for all $v\notin S_{\mathrm{fin}}$.

Proof.

Let $\overline{v}$ be a fixed prime in $\mathcal{O}_{\overline{K}}$ over $v$ and let $G_{\overline{v}}$ be the decomposition group of $\overline{v}$. We observe that the double coset space $G_{L}\backslash G_{K}/G_{\overline{v}}$ is in bijection with the set of primes $w|v$ of $\mathcal{O}_{L}$ via the map $G_{L}\gamma\,G_{\overline{v}}\mapsto\gamma(\overline{v})\cap L$. We have the following bijections:

[TABLE]

The second bijection follows by Diagram 6.5 in Lemma 6.5 and by Diagram 5.2 in Theorem 5.19, Diagram 6.7 in Theorem 6.12 and Conjectures 7.5 and 7.19 that are assumed in this section. We would like to point out that applying Diagrams 6.5, 5.2 and 6.7 we should consider all quotient groups in these diagrams in the form of right cosets.

By a slight abuse of notation, let $\langle f_{v}\rangle$ denote the subgroup generated by the Frobenius element $f_{v}$ in $G(K_{0}/K)$ in the middle term of (11.16) and let $\langle f_{v}\rangle$ denote the subgroup generated by the normalized Frobenius $f_{v}$ in $\mathcal{G}$ (denoted in the same way as Frobenius) in the right term of (11.16). Then the double coset space $\mathcal{H}\backslash\mathcal{G}/\langle f_{v}\rangle$ is in bijection with the set of primes $w|v$ of $\mathcal{O}_{L}$ via the map $\mathcal{H}\,\gamma\,\langle f_{v}\rangle\mapsto\gamma(\overline{v})\cap L$. Under this bijection, the decomposition group of $w$ over $v$ is the group $\gamma\langle f_{v}\rangle\gamma^{-1}/(\gamma\langle f_{v}\rangle\gamma^{-1}\cap\mathcal{H})$, so the inertia degree of $w$ over $v$ equals the index $[\gamma\langle f_{v}\rangle\gamma^{-1}:\gamma\langle f_{v}\rangle\gamma^{-1}\cap\mathcal{H}]$. Consequently, $w$ splits completely over $v$ if and only if $\gamma\langle f_{v}\rangle\gamma^{-1}\subseteq\mathcal{H}$, or equivalently if and only if $\gamma f_{v}\gamma^{-1}\in\mathcal{H}$.

To summarize, at this point we have a one-to-one correspondence between the double cosets $\mathcal{H}\gamma\langle f_{v}\rangle$ for which $\gamma f_{v}\gamma^{-1}\in\mathcal{H}$ and the set of primes $w|v$ of $\mathcal{O}_{L}$ that split completely over $v$. Now note that each such double coset consists of one right coset of $\mathcal{H}$:

[TABLE]

We may thus replace the double cosets in the correspondence by the right cosets $\mathcal{H}\gamma$ for which $\gamma f_{v}\gamma^{-1}\in\mathcal{H}$. Since $\mathcal{G}=\bigsqcup_{i=0}^{t}\mathcal{H}g_{i}^{-1}$ by (11.13) and (11.14), this yields the desired result. ∎

We can write $f_{v}=s_{v}\,u_{v}$ where $s_{v}\in\mathcal{G}$ is the semisimple part of $f_{v}$ and $u_{v}\in\mathcal{G}$ is the unipotent part of $f_{v}$. Recall that $s_{v}\,u_{v}=u_{v}\,s_{v}$.

Lemma 11.9**.**

By the isomorphism (11.14) and Lemma 11.4, there are natural bijections of double coset spaces:

[TABLE]

Proof.

Observe that (cf. the proof of Corollary 7.31)

[TABLE]

Observe that $\forall g\in\mathcal{G}$ and $\forall i$ we obtain $gu_{v}^{i}g^{-1}\in U(\mathcal{G})$. Hence by Lemma 11.4 and (11.18) we have:

[TABLE]

Hence:

[TABLE]

Therefore:

[TABLE]

Recall that $\langle s_{v}\rangle\subset G$. Assume that for $g_{1},g_{2}\in G$ the double cosets $Hg_{1}\langle s_{v}\rangle$ and $Hg_{2}\langle s_{v}\rangle$ in $H\backslash G/\langle s_{v}\rangle$ map to the same double coset $\mathcal{H}g_{1}\langle s_{v}\rangle=\mathcal{H}g_{2}\langle s_{v}\rangle$ in $\mathcal{H}\backslash\mathcal{G}/\langle s_{v}\rangle$. Hence $g_{2}\langle s_{v}\rangle{\langle s_{v}\rangle}^{-1}g_{1}^{-1}\subset\mathcal{H}$. Therefore $Hg_{1}\langle s_{v}\rangle=Hg_{2}\langle s_{v}\rangle$ by (11.15). It follows that:

[TABLE]

Now the Lemma follows by (11.21) and (11.22). ∎

Corollary 11.10**.**

There is a one-to-one correspondence between the set of elements $g_{i}\in G$ such that $g_{i}^{-1}s_{v}g_{i}\in H$ and the set of primes $w\,|\,v$ of $\mathcal{O}_{L}$ that split completely over $v$, for all $v\notin S_{\mathrm{fin}}$.

Proof.

By lemma 11.8 there is a one-to-one correspondence between the set of elements $g_{i}\in G$ such that $g_{i}^{-1}f_{v}g_{i}\in\mathcal{H}$ and the set of primes $w\,|\,v$ of $\mathcal{O}_{L}$ that split completely over $v$. On the other hand by Lemma 11.4, containment (11.18), equality (11.15) and Lemma 11.9 we have:

[TABLE]

For $m\in\mathbb{N}$ put $N_{K}(m):=\#\{v\in{\text{Spec}}\,\mathcal{O}_{K}\colon Nv\leq m\}$. Define $N_{L}(m)$ in the same way. E. Landau proved that $N_{K}(m)$ and $N_{L}(m)$ are asymptotically equal. Let $(w_{1})$ be the sequence of primes in ${\text{Spec}}\,\mathcal{O}_{L}$ that split completely over $K$ and let $S_{L}(m):=\#\{w_{1}\in{\text{Spec}}\,\mathcal{O}_{K}\colon Nw_{1}\leq m\}$. We observe that $S_{L}(m)$ and $N_{L}(m)$ are asymptotically equal (cf. the proof of Lemma 11.3).

Let $\chi={{\rm{Ind}}}_{H}^{G}\chi_{1}$. By Lemma 11.8 and (10.18) we obtain:

[TABLE]

By Lemma 10.14 and (11.23), the following equalities (11.24) and (11.25) are equivalent:

[TABLE]

Let $(s_{v})$ (resp. $(s_{w_{1}}))$ denote the sequence in $X(G)$ of conjugacy classes of elements $s_{v}$ (resp. sequence in $X(H)$ of conjugacy classes of elements $s_{w_{1}}$) by slight abuse of notation.

Remark 11.11*.*

Assume that the character $\chi$ in formula (10.25) is irreducible. Because $\chi$ and $\chi_{i,j}$ are irreducible, $\int_{G}\,{\chi}\,d\mu=0$ and $\int_{H_{i}}\,{\chi_{i,j}}\,d\mu_{H_{i}}=0$ for all $\chi_{i,j}\not=1$. Integrating (10.25) over $G$ and applying Lemma 10.14, we obtain:

[TABLE]

Theorem 11.12**.**

Under the assumptions of Theorem 7.17 (resp. 7.26 ), Sato–Tate Conjecture 7.12 (resp. Sato–Tate Conjecture 7.33) holds for the representation $\rho_{l}$ (resp. $\widetilde{\rho}_{l}$) with respect to $\operatorname{ST}_{K}(V,\psi)$ (resp. $\widetilde{\operatorname{ST}_{K}}(V,\psi)$) if and only if for each subextension $K_{1}$ of $K_{0}/K$ for which $K_{0}/K_{1}$ is cyclic, the same conjecture holds for the representation $\rho_{l}\,|\,G_{K_{1}}$ (resp. $\widetilde{\rho}_{l}\,|\,G_{K_{1}}$) with respect to $\operatorname{ST}_{K_{1}}(V,\psi)$ (resp. $\widetilde{\operatorname{ST}_{K_{1}}}(V,\psi)$).

Proof.

The “only if” direction is proven in the same way as Lemma 11.6. The ”if” direction is proven as follows. Let $H_{i}$ be the groups that appear in equations (10.22) and (10.23). By Proposition 7.15 (resp. Proposition 7.24) there is a field subextension $K\subset L_{i}\subset K_{0}$, for each $1\leq i\leq s$, such that $H_{i}=\operatorname{ST}_{L_{i}}(V,\psi)$ (resp. $H_{i}=\widetilde{\operatorname{ST}_{L_{i}}}(V,\psi)$). By hypothesis, the Sato–Tate conjecture holds for each group $H_{i}$. Then for each $i,j$ we can apply (11.25) to get:

[TABLE]

Let $\chi\in X(G)$ be irreducible and consider the formula (10.25). By (11.23), (11.26), and (11.27):

[TABLE]

Consequently $(s_{v})$ is equidistributed in $X(G)$. ∎

Remark 11.13*.*

A stronger form of Theorem 11.12, in which the Sato–Tate conjecture for $\operatorname{ST}_{K}(V,\psi)$ and $\operatorname{ST}_{K_{0}}(V,\psi)$ are equated, was stated in [BK2, Theorem 6.12] and was left without proof. We do not know how to prove such a statement in general. However, Theorem 10.20 implies that this holds when the derived group of $G_{0}$ has pairwise distinct simple factors, none of which admits a nontrivial diagonal automorphism (this condition rules out factors of types $A_{n},D_{4},E_{6}$).

12. Relations among motivic categories

Let $K$ be a field contained in $\mathbb{C}$. Let ${{\rm SmProj}_{K}}$ denote the category of smooth projective varieties over $K$ and let $\mathcal{V}$ be a full subcategory of ${{\rm SmProj}_{K}}$. Let $\mathcal{M}_{{\rm{\sim}}}:=\mathcal{M}_{K,{{\rm{{}_{\sim}}}}}$ denote the motivic category corresponding to an equivalence relation $\sim$. In our applications, $\sim$ will be one of the following properties defining the corresponding motivic category:

• $\sim\,={\rm{rat}}$

:  rational equivalence,

• $\sim\,={\rm{alg}}$

:  algebraic equivalence,

• $\sim\,={\rm{\otimes}}$

:  smash nilpotent equivalence,

• $\sim\,={\rm{hom}}$

:  homological equivalence,

• $\sim\,={\rm{num}}$

:  numerical equivalence,

• $\sim\,={\rm{mot}}$

:  motivated cycles of André,

• $\sim\,={\rm{ahc}}$

:  absolute Hodge cycles of Deligne.

We will also denote $\mathcal{M}_{\rm{hs}}:=\mathcal{M}_{\mathbb{Q},{\rm{hs}}}$ the category of $\mathbb{Q}$-rational Hodge structures. Y. André [An1, Chap. 9] constructs the category of motivated cycles $\mathcal{M}_{{\rm{mot}},\,\mathcal{V}}:=\mathcal{M}_{K,{{\rm{mot}}}}$ modeled on $\mathcal{V}$. We will denote $\mathcal{M}_{{\rm{mot}}}:=\mathcal{M}_{{\rm{mot}},\,{{\rm SmProj}_{K}}}$.

Diagram 12.1 indicates the functors relating these motivic categories (cf. [MNP, p. 30]; see also [An1] concerning the category $\mathcal{M}_{{\rm{mot}}})$.

In the construction of $\mathcal{M}_{\sim\,}$ (see for example [DM, p. 200], [An1, p. 31], [MNP, Chap. 2] etc.) one starts with the category ${{\rm SmProj}_{K}}$ and constructs the new category with objects $h_{{\sim}}(X)$ and morphisms between them:

[TABLE]

for $X$ and $Y$ smooth projective over $K$ and $X$ of pure dimension $d$. The $\mathbb{Q}$-vector space $C_{\sim}^{d}(X,Y)$ is defined as follows:

[TABLE]

where $\operatorname{CH}_{\operatorname{AH}}^{d}(X\times Y)$ is defined in [D1, p. 34] and $\operatorname{CH}_{{\rm{mot}}}^{d}(X\times Y)$ is the image of the natural map $Z^{d}_{{\rm{mot}}}(X\times Y)\rightarrow H^{2d}(X\times Y)(d)$ (see [An1, pp 95–96]). For any smooth, projective $X$ over $K$, the motivated cycles $Z^{\ast}_{{\rm{mot}}}(X)$ do not depend on the choice of a Weil cohomology $H^{\ast}$ up to natural isomorphism [An1, Lemme 9.2.1.2].

Based on this, one performs the classical construction due to Grothendieck to obtain corresponding categories of motives $\mathcal{M}_{{\sim}}$ (see for example [An1, p.31–35], [DM, pp. 200–205], [MNP, p. 25–29]).

13. Assumptions on $\mathcal{M}_{{\sim}}$

From now on in this paper, we will work under the following classical assumptions on the category $\mathcal{M}_{{\sim}}$.

ASSUMPTION 1.   The Chow–Künneth decomposition holds in $\mathcal{M}_{{\sim}}$: there are idempotents $p_{r}(X)\in{\rm{End}}_{\mathcal{M}_{\sim}}(h_{{\sim}}(X))$ orthogonal to each other lifting the Künneth components in Betti, étale, and de Rham realizations such that:

[TABLE]

where $h_{{\sim}}^{r}(X)=(h_{{\sim}}(X),p_{r}(X),0)$.

Remark 13.1*.*

Observe that the Chow–Künneth decomposition holds in $\mathcal{M}_{{\rm{ahc}}}$ (see [DM, p. 202]) and in $\mathcal{M}_{{\rm{mot}}}$ (see [An2, Prop. 2.2]; cf. [An1, Lemme 9.2.1.3]).

Remark 13.2*.*

The Chow–Künneth conjecture for rational equivalence (see [MNP, Chapter 6, pp. 67–69]) states that there is a Chow–Künneth decomposition:

[TABLE]

The Chow–Künneth conjecture directly implies the Chow–Künneth decomposition for all equivalence relations ${\sim}$ listed above except ${\sim}={\rm{num}}$. Note that Assumption 1 is not well formulated for ${\sim}={\rm{num}}$ because the realization functors are not available (see [MNP, §2.6]). If we assume Standard Conjecture D (equivalence between homological and numerical equivalence relations), then Assumption 1 for numerical equivalence becomes well formulated and equivalent to Assumption 1 for homological equivalence.

Definition 13.3**.**

Let $h_{{\sim}}(X)(m)$ be the twist of $h_{{\sim}}(X)$ by the $m$-th power of the Lefschetz motive $\mathbb{L}:=h_{{\sim}}^{2}(\mathbb{P}^{1})$ for some $m\in\mathbb{Z}$. A direct summand of a motive of the form $h_{{\sim}}^{r}(X)(m)$ will be called a homogeneous motive of weight $n:=2m-r$.

Recall that ${\bf{1}}:=h_{{\sim}}({\rm{Spec}}\,K)$. Hence $\mathbb{L}={\bf{1}}(-1)$ so $\mathbb{L}^{\otimes\,r}={\bf{1}}(-r)$.

Remark 13.4*.*

Let $L/K$ be a field extension such that $K\subset L\subset\overline{K}$. If $M\in\mathcal{M}_{{\sim}}$ is a direct summand of $h_{{\sim}}(X)(m)$, then $M\otimes_{K}L$ is a direct summand of $h_{{\sim}}(X\otimes_{K}L)(m)$ in $\mathcal{M}_{L,{\sim}}\,\,$.

The properties of the Chow–Künneth decomposition (cf. [MNP, Def. 6.1.2 and Example 6.1.5. (3)]), the properties of Weil cohomology, and the cycle map [Kl, p. 10] imply the existence of (Motivic Poincaré duality)

[TABLE]

whose Betti, étale, and de Rham realizations give corresponding Poincaré dualities in Betti, étale, and de Rham cohomologies.

ASSUMPTION 2.   The category $\mathcal{M}_{{\rm{\sim}}}$ is Tannakian and semisimple over $\mathbb{Q}$.

Let $H_{B}$ denote now the corresponding fiber functor:

[TABLE]

Remark 13.5*.*

• (1)

The category $\mathcal{M}_{{\rm{ahc}}}$ is Tannakian and semisimple over $\mathbb{Q}$ [DM, Prop. 6.5].

• (2)

For every motive $M$ in $\mathcal{M}_{{\rm{mot}}}$, for any “suitably large” (depending on $M$) family of absolute cycles $\mathcal{V}$, the tensor subcategory of $\mathcal{M}_{{\rm{mot}},\mathcal{V}}$ generated by $M$ is Tannakian and semisimple [An1, Théorème 9.2.3.1]. Consequently, $\mathcal{M}_{{\rm{mot}}}$ is Tannakian and semisimple.

• (3)

For the category $\mathcal{M}_{{\rm{hom}}}$ the Betti realization $H_{B}$ is a fiber functor. For $\mathcal{M}_{{\rm{num}}}$ the Betti realization $H_{B}$ is not well defined in general. Under Standard Conjecture D (equivalence of $\mathcal{M}_{{\rm{num}}}$ with $\mathcal{M}_{{\rm{hom}}}$), $H_{B}$ is well defined on $\mathcal{M}_{{\rm{num}}}$. Moreover Standard Conjecture D implies Standard Conjecture C (i.e. $p_{r}(X)\in{\rm{End}}_{\mathcal{M}_{{\rm{hom}}}}(h_{{\rm{hom}}}(X))$), see [An1, Théorème 5.4.2.1].

• (4)

Jannsen proved [Ja2] that $\mathcal{M}_{{\rm{num}}}$ is a semisimple abelian category. He also proved [Ja2] that under Conjecture C, the category $\mathcal{M}_{{\rm{num}}}$ is Tannakian and semisimple. Hence under Standard Conjecture D, both categories $\mathcal{M}_{{\rm{hom}}}$ and $\mathcal{M}_{{\rm{num}}}$ are Tannakian and semisimple.

Next we want an assumption that provides a polarization on the homogeneous motives $h^{r}_{{\sim}}(X)(m)$ in $\mathcal{M}_{{\sim}}$.

ASSUMPTION 3. There is a natural morphism (Motivic star operator) for every smooth projective $X$:

[TABLE]

sent via Betti, étale, and de Rham realizations onto the morphism $H^{r}(X)\rightarrow H^{2d-r}(X)$, which is defined by an absolute Hodge cycle (see [DM, Proposition 6.1 and discussion on p. 199]).

Remark 13.6*.*

Assumption 3 leads to the following natural morphism in $\mathcal{M}_{{\sim}}$:

[TABLE]

Under Assumption 3 the functor $\mathcal{M}_{{\sim}}\rightarrow\mathcal{M}_{{{\rm{ahc}}}}$ sends $\psi_{\sim}^{r}$ to $\psi^{r}$ [DM, p. 199]:

[TABLE]

Assumption 3 holds for $\mathcal{M}_{{{\rm{ahc}}}}$ (see loc. cit.). Hence for $h_{{\sim}}^{r}(X)$ in $\mathcal{M}_{{\sim}}$, we obtain a polarization of the real Hodge structure associated with the rational Hodge structure on the Betti realization $H_{B}(h_{{\sim}}^{r}(X))\,=\,H^{r}(X(\mathbb{C}),\,\mathbb{Q})$, because $H_{B}$ factors through $\mathcal{M}_{{{\rm{ahc}}}}$ and $\psi^{r}$ gives the polarization on $\mathcal{M}_{{{\rm{ahc}}}}$ [DM, pp. 197–199] (cf. [Pan, p. 478–480], [Ja1, pp. 2–4]). By taking a Tate twist in (13.4) and applying the Betti realization, we also obtain a polarization on $h_{{\sim}}^{r}(X)(m)$. Therefore, for any homogeneous motive $M\in\mathcal{M}_{{{\sim}}}$ of weight $n$, there is the following map in $\mathcal{M}_{{\sim}}$:

[TABLE]

By [PS, Cor. 2.12, p. 40] we obtain the polarization of the real Hodge structure associated with rational Hodge structure on the Betti realization $V:=H_{B}(M)$:

[TABLE]

Concerning the polarization for $\mathcal{M}_{{\rm{mot}}}$, it follows from [An2, Prop. 2.2, p. 16], (cf. [An1, Remarque 9.2.2.1]) that for $M=h^{r}(X)\in\mathcal{M}_{{\rm{mot}}}$, the element $\psi^{r}\in\operatorname{CH}^{2d-r}_{\operatorname{AH}}(X\times X)$ discussed above comes from a motivated cycle. Hence as above, for the case of a homogeneous motive $M$ in $\mathcal{M}_{{\rm{ahc}}}$, we can associate with $M$ a polarization of the real Hodge structure associated with the rational Hodge structure on the Betti realization $V=H_{B}(M)$. Consequently, Assumption 3 also holds for $\mathcal{M}_{{\rm{mot}}}$.

Concerning polarizations for all other $\mathcal{M}_{\sim}$, we observe that if $\psi^{r}\in\operatorname{CH}^{2d-r}_{\operatorname{AH}}(X\times X)$ comes from an algebraic cycle, then we would have the natural Hodge structure as in $\mathcal{M}_{{\rm{ahc}}}$ and $\mathcal{M}_{{\rm{mot}}}$.

Let $\overline{\mathcal{M}}_{\sim}:={\mathcal{M}_{\overline{K},\sim}}$ and let

[TABLE]

Because $\overline{\mathcal{M}}_{\sim}$ is semisimple, $D$ is a finite dimensional $\mathbb{Q}$-vector space. Because $\overline{\mathcal{M}}_{\sim}$ is Tannakian and $V=H_{B}(M)$, $D\subset\operatorname{End}_{\mathbb{Q}}(V)$. Hence in this case we obtain corresponding representation $\rho_{e}$, $K_{e}:=\overline{K}^{\operatorname{Ker}\rho_{e}}$ and the twisted decomposable algebraic Lefschetz group $\operatorname{DL}_{K}(V,\psi,D)$ (see (2.10), (2.13)).

ASSUMPTION 4. For every two objects $M,N$ of $\mathcal{M}_{{\rm{\sim}}}$ one has:

[TABLE]

Remark 13.7*.*

The property (13.9) holds in $\mathcal{M}_{{\rm{ahc}}}$ by [DM, p. 215], [Ja1, p. 53] and in $\mathcal{M}_{{\rm{mot}}}$ by [An2, Scolie 2.5], [An2, Sec. 4].

Remark 13.8*.*

The Artin motive of principal interest in this paper is ${\rm{Hom}}_{\overline{\mathcal{M}}_{{\rm{\sim}}}}(\overline{M},\,\overline{N})$. The subcategory of Artin motives in every $\mathcal{M}_{{\rm{\sim}}}$ is the same regardless of what $\sim$ is (cf. [An1, p. 35]). So the category of Artin motives will be always denoted $\mathcal{M}_{K}^{0}$.

Let $M$ be a homogeneous motive which is a direct summand of $h_{{\sim}}^{r}(X)(m)$.

Let $H^{r}:=H^{r}(X(\mathbb{C}),\,\mathbb{Q}(m))$ denote the Betti realization of $h_{{\sim}}^{r}(X)(m)$. The polarization of the rational Hodge structure on $H^{r}$ gives the rational, polarized Hodge structure $(V,\psi)$, where $V=H_{B}(M)$ (see (13.7)).

Let $D$ be the $\mathbb{Q}$-algebra defined in (13.8).

Lemma 13.9**.**

The Hodge structure $(V,\psi)$ and the $\mathbb{Q}$-algebra $D$ associated with the motive $M$ satisfy conditions (D1), (D2) of §2.

Proof.

By the assumption the category $\mathcal{M}_{{\rm{\sim}}}$ is Tannakian, hence the Betti realization is a fiber functor. This functor factors through $\mathcal{M}_{{\rm{ahc}}}$. Hence the condition (D1) holds by [DM, Prop. 6.1]. For the condition (D2) cf. [D1, Prop. 2.9]. ∎

Let $H^{r}_{{}_{\rm{DR}}}:=H^{r}_{{}_{\rm{DR}}}(X)(m)$ denote the De Rham realization of $h_{{\sim}}^{r}(X)(m)$ and let $V_{{}_{\rm{DR}}}$ be the De Rham realization of $M$. The De Rham realization of the map (13.6) gives the space $(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$.

Lemma 13.10**.**

The space $(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$ and the $\mathbb{Q}$-algebra $D$ associated with the motive $M$ satisfy conditions (DR1), (DR2) of §3.

Proof.

(DR1) follows because the De Rham realization of $\mathcal{M}_{{\rm{\sim}}}$ is also a fiber functor by comparison with the Betti realization. (DR2) follows by compatibility of the Hodge decompositions of $V_{{}_{\rm{DR}}}\otimes_{K}\mathbb{C}$ and $V\otimes_{\mathbb{Q}}\mathbb{C}$ and by Lemma 13.9. ∎

Let $H^{r}_{l}:=H^{r}_{et}({\overline{X}},\,\mathbb{Q}_{l}(m))$ be the $l$-adic realization of $h_{{\sim}}^{r}(X)(m)$ and let $V_{l}$ denote the $l$-adic realization of $M$. Consider the natural representations $\rho_{H^{r}_{l}}\colon G_{K}\rightarrow\operatorname{GIso}_{(H^{r}_{l},\psi_{l})}(\mathbb{Q}_{l})$ and $\rho_{l}\colon G_{K}\rightarrow\operatorname{GIso}_{(V_{l},\psi_{l})}(\mathbb{Q}_{l})$.

Remark 13.11*.*

It is not known in general that the family of $l$-adic representations attached to a motive in the motivic category $\mathcal{M}_{{\rm{\sim}}}$ is strictly compatible in the sense of Serre. In particular it is the case in $\mathcal{M}_{{\rm{ahc}}}$ [Pan, p. 475] and in $\mathcal{M}_{{\rm{mot}}}$ [An1, p. 96]. However for the motive $h^{r}_{{\sim}}(X)(m)$ the corresponding family of $l$-adic representations is strictly compatible. For a homogeneous motive $M$ the $l$-adic realizations would be a strictly compatible family if, for example, the idempotents which cut $M$ out of $h^{r}_{{\sim}}(X)(m)$ come from algebraic cycles.

Lemma 13.12**.**

If the Hodge structure $(V,\psi)$, the $\mathbb{Q}$-algebra $D$, and the representation $\rho_{l}$ for the motive $M$ are such that the family $(\rho_{l})$ is strictly compatible, then conditions (R1)–(R4)* of §4 are satisfied.*

Proof.

The condition (R1) holds because for every prime ideal $\lambda\,|\,l$ in $\mathcal{O}_{K}$ the category of $G_{K_{\lambda}}$-Hodge–Tate modules is a Tannakian tensor category over $\mathbb{Q}_{l}$ [Se4, p. 157]. To check condition (R2), take a big enough set $S$ of primes of $\mathcal{O}_{K}$, including primes over $l$, such that $X$ has a proper and regular model $\mathcal{X}_{S}$ over $\mathcal{O}_{K,S}$. It follows from the smooth proper base change theorem [Mi1, Chap. VI, Cor. 4.2] that for every prime $v\in{\rm{Spec}}\,\mathcal{O}_{K,S}$, writing $\mathcal{X}_{v}$ for the special fiber over the residue field $k_{v}$, there is a canonical isomorphism of $G(\overline{k_{v}}/k_{v})$-modules:

[TABLE]

Hence, by Deligne’s proof of the Weil conjectures [D3], the Frobenius action on $W_{l}$ has eigenvalues of $\mathbb{C}$-absolute value $q_{v}^{-\frac{n}{2}}$. Hence the same follows for the action of Frobenius on $V_{l}$. Condition (R3) holds for the motive $h^{i}_{{\sim}}(X)(m)$ by Remark 4.3 and for the motive $M$ by construction of a motivic category, its realizations and filtration preserving the comparison isomorphisms between Betti, de Rham, and étale cohomologies. Condition (R4) is obvious. ∎

14. Motivic Galois group and motivic Serre group

From now on we will use the following notation (cf. [BK2, Chapter 9]):

• $\bullet$

$\mathcal{M}_{{\sim}}(M)$ denotes the smallest strictly full Tannakian subcategory of $\mathcal{M}_{{\sim}}$ containing the object $M$ of $\mathcal{M}_{{\rm{\sim}}}$,

• $\bullet$

$G_{\mathcal{M}_{{\sim}}}:={\rm Aut}^{\otimes}(H_{B})$ is the motivic Galois group for $\mathcal{M}_{{\sim}}$,

• $\bullet$

$G_{\mathcal{M}_{{\sim}}(M)}:={\rm Aut}^{\otimes}(H_{B}|\mathcal{M}_{{\sim}}(M))$ is the motivic Galois group for $\mathcal{M}_{{\sim}}(M)$,

• $\bullet$

$h^{0}(D)$ denotes the Artin motive corresponding to $D$,

• $\bullet$

$\mathcal{M}_{K}^{0}(D)$ is the smallest Tannakian subcategory of $\mathcal{M}_{K}^{0}$ containing $h^{0}(D)$,

• $\bullet$

$G_{\mathcal{M}_{K}^{0}(D)}:={\rm Aut}^{\otimes}(H_{B}|\mathcal{M}_{K}^{0}(D))$ is the motivic Galois group for $\mathcal{M}_{K}^{0}(D)$.

It is well known that $\mathcal{M}_{K}^{0}$ is a semisimple Tannakian category. Hence $\mathcal{M}_{K}^{0}(D)$, being a strictly full Tannakian subcategory, is also semisimple. Similarly, by assumption, $\mathcal{M}_{{\sim}}$ is semisimple, so the strictly full Tannakian subcategory $\mathcal{M}_{{\sim}}(M)$ is also semisimple. Hence the algebraic groups $G_{\mathcal{M}_{\sim}(M)}$ are reductive but not necessarily connected (see also [DM, Prop. 2.23, p. 141], [DM, Prop. 6.23, p. 214], [Se5, p. 379]).

By [DM, Prop. 6.1 (e), p. 197] the Hodge decomposition of $V\otimes_{\mathbb{Q}}\mathbb{C}$ is $D=D(M)$-equivariant for $M=h_{{\sim}}^{r}(X)$. Hence for any homogeneous motive $M\in\mathcal{M}_{{\sim}}$, the Betti realization $V:=H_{B}(M)$ admits a Hodge decomposition of $V\otimes_{\mathbb{Q}}\mathbb{C}$ which is $D=D(M)$-equivariant (cf. [PS, Cor. 2.12, p. 40]).

In the same way as in [BK2], we have the following properties of the motivic categories $\mathcal{M}_{{\sim}}$, $\mathcal{M}_{{\sim}}(M)$, and $\mathcal{M}_{K}^{0}(D)$. By the assumption (13.9):

[TABLE]

Hence the motive $h^{0}(D)$ splits off of $\underline{{\rm End}}_{\mathcal{M}_{\sim}(M)}(M)=M^{\vee}\otimes M$ in $\mathcal{M}_{\sim}(M)$. Since $G_{\mathcal{M}_{K}^{0}}\simeq G_{K}$, we observe that

[TABLE]

Hence the top horizontal and left vertical maps in the following Diagram 14.1 are faithfully flat (see [DM, (2.29)]):

In particular all homomorphisms in Diagram 14.1 are surjective.

By (13.6), (13.7), and the definition and properties of ${\rm Aut}^{\otimes}(H_{B}|\mathcal{M}_{\sim}(M))$ cf. [DM, p. 128–130], we obtain:

[TABLE]

Definition 14.1**.**

Define the following algebraic groups:

[TABLE]

The algebraic group $G_{\mathcal{M}_{\sim}(M),1}$ will be called the motivic Serre group.

Definition 14.2**.**

For any $\tau\in\operatorname{Gal}(K_{e}/K)$, put

[TABLE]

Remark 14.3*.*

The bottom horizontal arrow in the diagram (14.1) is

[TABLE]

Let $g\in G_{\mathcal{M}_{\sim}(M)}$ and let $\tau:=\tau(g)$ be the image of $g$ via the map (14.5). Hence for any element $\beta\in D$ considered as an endomorphism of $V$ we have:

[TABLE]

It follows from (14.3), the surjectivity of (14.5), and (14.6) that:

[TABLE]

Because

[TABLE]

we have:

[TABLE]

The map (14.5) gives the following natural map:

[TABLE]

Let $\tau\in\operatorname{Gal}(K_{e}/K)$. By (2.12), (2.13), (14.4), (14.10), and Definition 14.1 we obtain:

[TABLE]

15. Motivic Mumford–Tate group and motivic Serre group

Because $X/K$ is smooth projective, Remarks 4.1 and 4.6 show that the $l$-adic realization $V_{l}:=H^{r}(\overline{X},\,\mathbb{Q}_{l}(m))$ of the motive $h_{{\sim}}^{r}(X)(m)$ is of Hodge–Tate type. Hence Bogomolov’s theorem applies, so the image of the representation $\rho_{l}$ contains an open subgroup of homotheties of the group $\operatorname{GL}(V_{l})$ [Su, Prop. 2.8], provided $H^{r}(\overline{X},\,\mathbb{Q}_{l}(m))$ has nonzero weight. The nonzero weight assumption is essential because for $X$ of dimension $d$, the $G_{K}$-module $H^{2d}(\overline{X},\,\mathbb{Q}_{l}(d))\simeq\mathbb{Q}_{l}$ is trivial, and hence the image of the Galois representation does not contain nontrivial homotheties.

From now on, let $M\in\mathcal{M}_{{\sim}}$ be a homogeneous motive, that is, a direct summand of a motive $h^{r}_{{\sim}}(X)(m)$ for some $r$ and $m$. We assume that the $l$-adic realization of $h^{r}_{{\sim}}(X)(m)$ has nonzero weights with respect to the $G_{K}$-action. The $l$-adic realization of $\overline{M}$ is a $\mathbb{Q}_{l}[G_{F}]$-direct summand of the $l$-adic realization of $h^{r}_{{\sim}}(X)(m)$. Hence the $l$-adic representation corresponding to $V_{l}:=H_{l}(\overline{M})$ has image that contains an open subgroup of homotheties.

Remark 15.1*.*

The $l$-adic representation

[TABLE]

associated with $M$ factors through $G_{\mathcal{M}_{\sim}(M)}(\mathbb{Q}_{l})$ by the definition of $G_{\mathcal{M}_{\sim}(M)}$ and the comparison isomorphism of Betti and étale realizations $H_{B}(M)\otimes_{\mathbb{Q}}\,\mathbb{Q}_{l}\,\simeq\,H_{et}(\overline{M})$ (cf. [Se5, p. 386]). Hence

[TABLE]

where ${G_{\mathcal{M}_{\sim}(M)}}_{\mathbb{Q}_{l}}:={G_{\mathcal{M}_{\sim}(M)}}\otimes_{\mathbb{Q}}\mathbb{Q}_{l}$.

The following commutative Diagram 15.1 follows from the definitions of the corresponding group schemes and (15.2). All horizontal arrows are closed immersions and the columns are exact.

In particular:

[TABLE]

The following theorem extends [BK2, Theorem 10.2].

Theorem 15.2**.**

The group scheme $G_{\mathcal{M}_{{\sim}}(M),1}$ is reductive. There is a short exact sequence of finite groups:

[TABLE]

In particular $G_{\mathcal{M}_{{\sim}}(M),1}^{0}$ is connected if and only if $i_{M}$ is an isomorphism.

Proof.

We will write $\mathcal{M}(M)$ for $\mathcal{M}_{{\sim}}(M)$ in the following commutative Diagram 15.2 to make notation simpler.

By our assumptions $\mathcal{M}_{{\sim}}$ is semisimple. The category $\mathcal{M}_{{\sim}}(M)$ is a strictly full subcategory of $\mathcal{M}_{{\sim}}$ because by Tannaka duality (Theorem 9.2), these categories are equivalent to the corresponding categories of representations of $G_{\mathcal{M}_{{\sim}}}$ and $G_{\mathcal{M}_{{\sim}}(M)}$. Hence $\mathcal{M}_{{\sim}}(M)$ is semisimple and consequently $G_{\mathcal{M}_{{\sim}}(M)}$ is reductive. The exactness of the middle vertical column of Diagram 15.2 follows by the definition of $G_{\mathcal{M}_{{\sim}}(M),1}$ and the exactness of the middle column in Diagram 15.1. This shows that $G_{\mathcal{M}_{{\sim}}(M),1}$ is also reductive. Observe that the left vertical column is not a priori exact at the term $(G_{\mathcal{M}(M)})^{\circ}$. By definition the rows of Diagram 15.2 are exact. Hence the map $i_{M}$ is surjective. By the definition of $G_{\mathcal{M}_{{\sim}}(M),1}^{0}$ (see Definition 14.1) the right vertical column is also exact. Hence (15.4) is exact. Because $G_{\mathcal{M}_{{\sim}}(M),1}^{0}$ has the same dimension as $G_{\mathcal{M}_{{\sim}}(M),1}$, we obtain by (15.4) that $G_{\mathcal{M}_{{\sim}}(M),1}^{0}$ is connected if and only if $G_{\mathcal{M}_{{\sim}}(M),1}^{0}=(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}$ if and only if $i_{M}$ is an isomorphism if and only if the left column of Diagram 15.2 is exact. ∎

Corollary 15.3**.**

There are natural isomorphisms:

[TABLE]

In particular the natural map (14.11) is surjective.

Proof.

The isomorphism (15.5) follows from the surjectivity of $i_{M}$ in (15.4) of Theorem 15.2 and from (14.9). The isomorphism (15.6) follows from (2.14), (14.8), (14.12), (14.13), and (15.5). ∎

Corollary 15.4**.**

The group $G_{\mathcal{M}_{{\sim}}(M),1}^{0}$ is connected if and only if

[TABLE]

Proof.

This follows from Theorem 15.2 and (15.5). ∎

Lemma 15.5**.**

We have the following inclusions:

[TABLE]

Proof.

Algebraic cycles and motivated cycles are absolute Hodge cycles in cohomological realization [D1, Example 2.1 (a)], [An1, Remarque 9.2.2.1]. Hence these three types of cycles are Hodge cycles in cohomological realization. Moreover both groups in (15.8) are reductive. Now applying the “tensor invariance criterion” for reductive groups, we obtain (15.8). Moreover by (15.8) we obtain the following commutative Diagram 15.3 in which all horizontal arrows are closed immersions, the columns are exact, and the top commutative squares are cartesian.

The property (15.10) follows immediately from Diagram 15.3. ∎

The following Proposition is the motivic analogue of Proposition 2.1.

Proposition 15.6**.**

The following statements hold:

• (a)

$\mathbb{G}_{m}{\rm{Id}}_{V}\cdot(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}\,=\,\mathbb{G}_{m}{\rm{Id}}_{V}\cdot G_{\mathcal{M}_{{\sim}}(M),1}^{0}\,=\,(G_{\mathcal{M}_{{\sim}}(M)})^{\circ}$.

• (b)

$G_{\mathcal{M}_{{\sim}}(M),1}^{0}=(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}\,\cup\,-{\rm{Id}}_{V}\cdot(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}$.

• (c)

$-{\rm{Id}}_{V}\in(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}$* iff $(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}=G_{\mathcal{M}_{{\sim}}(M),1}^{0}$.*

• (d)

When $n$ is odd then $-{\rm{Id}}_{V}\,\in(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}=G_{\mathcal{M}_{{\sim}}(M),1}^{0}$.

Proof.

(a) Consider a coset $g(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}$ in $G_{\mathcal{M}_{{\sim}}(M),1}^{0}$. Applying [Hu, Section 7.4, Prop. B(b)] to the homomorphism

[TABLE]

we observe that $\mathbb{G}_{m}{\rm{Id}}_{V}\cdot(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}$ and $\mathbb{G}_{m}{\rm{Id}}_{V}\cdot g(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}$ are closed in $(G_{\mathcal{M}_{{\sim}}(M)})^{\circ}$. They are also of the same dimension as $(G_{\mathcal{M}_{{\sim}}(M)})^{\circ}$ because of the exact sequence

[TABLE]

and the finiteness of $G^{0}_{\mathcal{M}_{{\sim}}(M),1}/(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}$.

Because $(G_{\mathcal{M}_{{\sim}}(M)})^{\circ}$ is an irreducible algebraic group and $\mathbb{G}_{m}{\rm{Id}}_{V}\cdot(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}$ is a closed subgroup of the same dimension, we must have

[TABLE]

(b) From (15.12) there are $\alpha,\beta\in\mathbb{G}_{m}$ and $g_{1},g_{2}\in(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}$ such that

[TABLE]

Applying $\chi$ to (15.13) we obtain $\alpha^{2}=\beta^{2}$. This implies that $\pm{\rm{Id}}_{V}\cdot g_{1}=gg_{2}$. Hence $g(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}=(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}$ or $g(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}=-{\rm{Id}}_{V}(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}$.

(c) This follows immediately from (b).

(d) This follows immediately from (c) and the following Lemma 15.7. ∎

Lemma 15.7**.**

For odd weight $n$ we have:

[TABLE]

Proof.

The proof is basically the same as the proof of Lemma 2.3. We explain the modification. Extending the codomain of $\mu_{\infty,V}$ and $w$ of the proof of Lemma 2.3 using (15.8), we obtain the following corresponding cocharacters:

[TABLE]

These cocharacters have precisely the same properties as $\mu_{\infty,V}$ and $w$ of Lemma 2.3, as is clear from the bottom squares of Diagram 15.3. This leads directly to the cocharacter

[TABLE]

which splits $\chi$ in the following exact sequence:

[TABLE]

The group $(G_{\mathcal{M}_{{\sim}}(M)})^{\circ}$ is obviously connected. It follows by Lemma 2.2 that $G_{\mathcal{M}_{{\sim}}(M),1}^{0}$ is connected. Hence $G_{\mathcal{M}_{{\sim}}(M),1}^{0}=(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}$ because $(G^{0}_{\mathcal{M}_{{\sim}}(M),1})^{\circ}=(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}$ by Theorem 15.2. ∎

Definition 15.8**.**

The algebraic groups:

[TABLE]

will be called the motivic Mumford–Tate group and (as before) the motivic Serre group for $M$ respectively.

Serre made the following conjecture [Se5, sec. 3.4].

Conjecture 15.9**.**

(Serre)**

[TABLE]

Remark 15.10*.*

By Proposition 2.1(b), Lemma 15.5, and Proposition 15.6(b) and the diagram above it, Conjecture 15.9 holds for $M$ if and only if

[TABLE]

if and only if

[TABLE]

Definition 15.11**.**

A motive $M\in\mathcal{M}_{{\text{ahc}}}$ will be called an AHC motive if every Hodge cycle on any object of $\mathcal{M}_{{\text{ahc}}}(M)$ is an absolute Hodge cycle (cf. [D1, p. 29], [Pan, p. 473]).

Remark 15.12*.*

By [DM] Conjecture 15.9 holds for motives $h^{1}_{{\text{ahc}}}(A)$ where $A/K$ is an abelian variety and for AHC motives $M$ in $\mathcal{M}_{{\text{ahc}}}$ (cf. [Pan, Corollary p. 474]).

Similarly to [BK2, Chapter 10] we make the following definitions and state the following conjectures in this more general context of motivic categories.

Conjecture 15.13**.**

(Motivic Mumford–Tate) For any prime number $l$,

[TABLE]

By the diagram above Theorem 15.2, Conjecture 15.13 is equivalent to the following.

Conjecture 15.14**.**

(Motivic Sato–Tate) For any prime number $l$,

[TABLE]

Remark 15.15*.*

Conjecture 15.13 is equivalent to the conjunction of the following equalities:

[TABLE]

Similarly, Conjecture 15.14 is equivalent to the conjunction of the following equalities:

[TABLE]

Definition 15.16**.**

Let $M$ be a motive in $\mathcal{M}_{{\sim}}$. The Serre motivic parity group:

[TABLE]

is the component group of $G_{\mathcal{M}_{{\sim}}(M),1}^{0}$.

Proposition 15.17**.**

If $n$ is odd or if $(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}=C_{D}(\operatorname{Iso}_{(V,\psi)})$ (in particular if $\operatorname{H}(V,\psi)=C_{D}(\operatorname{Iso}_{(V,\psi)})$) then $\operatorname{P}_{S}(M,\psi_{{\sim}})$ is trivial.

Proof.

This follows by Lemma 15.7 and the following computation (apply (14.12) and Lemma 15.5):

[TABLE]

Proposition 15.18**.**

If $n$ is even and ${\rm dim}_{\mathbb{Q}}\,V$ is odd, then $\operatorname{P}_{S}(M,\psi_{{\sim}})$ is nontrivial.

Proof.

Again because $n$ is even, $\operatorname{Iso}_{(V,\psi)}=\operatorname{O}_{(V,\psi)}$. Hence (cf. Definition 14.1) we obtain $(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}\subset\operatorname{Iso}_{(V,\psi)}\subset{\pm{\rm{Id}}_{V}}\operatorname{SL}_{V}$. It follows that $-{\rm{Id}}_{V}\notin(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}$ because ${\rm{det}}(-{\rm{Id}}_{V})=(-1)^{{\rm{dim}}_{\mathbb{Q}}\,V}=-1$ and $(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}$ is connected. ∎

Example 15.19**.**

Let $A$ be an abelian threefold over $K$. The motive $h_{{\sim}}^{2}(A)$ has Betti realization $V:=H^{2}(A(\mathbb{C}),\,\mathbb{Q})=\bigwedge^{2}\,H^{1}(A(\mathbb{C}),\,\mathbb{Q})$. Hence by Example 2.15 and Proposition 15.18, the group $\operatorname{P}_{S}(h_{{\sim}}^{2}(A),\,\psi_{{\sim}})$ is nontrivial.

Example 15.20**.**

Let $A$ be an elliptic curve over $K$. The motive $\operatorname{Sym}^{2}\,h_{{\sim}}^{1}(A)$ has Betti realization $V:=\operatorname{Sym}^{2}\,H^{1}(A(\mathbb{C}),\,\mathbb{Q})$. Hence by Example 2.16 and Proposition 15.18, the group $\operatorname{P}_{S}(\operatorname{Sym}^{2}\,h_{{\sim}}^{1}(A),\,\psi_{{\sim}})$ is nontrivial.

Let $M$ be a homogeneous motive in $\mathcal{M}_{{\sim}}$. By (15.9) and (15.10) there is a natural epimorphism:

[TABLE]

By Corollary 3.8 we have a natural epimorphism:

[TABLE]

Hence by (15.22) we have a natural epimorphism:

[TABLE]

By (15.2) we have:

[TABLE]

Hence by (15.2) we obtain:

[TABLE]

It follows by (15.3) that there is a natural epimorphism:

[TABLE]

Remark 15.21*.*

Recall that each parity group has order at most 2. These groups are generated by corresponding cosets of $-{\rm{Id}}$. Moreover each homomorphism between parity groups carries the coset of $-{\rm{Id}}$ in the source into the coset of $-{\rm{Id}}$ in the target. Hence it is clear that each homomorphism between parity groups is an epimorphism.

Corollary 15.22**.**

If $\operatorname{P}_{S}(M,\psi_{{\sim}})$ is nontrivial then it is naturally isomorphic to each of the following parity groups: $\operatorname{P}(V,\psi)$, $\operatorname{P}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}})$, and $\operatorname{P}_{\rm{S}}(V_{l},\psi_{l})$.

Proof.

Every parity group is either trivial or isomorphic to $\{\pm 1\}$. Hence the corollary follows by epimorphisms (15.22), (3.18), (15.23), (15.24). ∎

Proposition 15.23**.**

Consider parity groups: $\operatorname{P}(V,\psi),\operatorname{P}(V_{{}_{\rm{DR}}},\psi_{{}_{\rm{DR}}}),\operatorname{P}_{\rm{S}}(V_{l},\psi_{l})$ and $\operatorname{P}_{S}(M,\psi_{{\sim}})$.

• (a)

If $n$ is odd, then all these groups are trivial.

• (b)

If $n$ is even and $\dim_{\mathbb{Q}}\,V$ is odd, then all these groups are nontrivial.

Proof.

(a) This follows from Propositions 2.13, 3.7, 5.14, 15.17.

(b) This follows from Propositions 2.14, 5.15, 15.18, and Corollary 3.9. ∎

16. The algebraic Sato–Tate group for motives

As in the previous section, we work with motives $M$ which are direct summands of motives of the form $h^{r}(X)(m)$; in this section, we propose a candidate for the algebraic Sato–Tate group for such motives. We prove, under the assumption in Definition 16.7, that our candidate for the algebraic Sato–Tate group is the expected one. In particular the assumption of Definition 16.7 holds if $M$ is an AHC motive (see Definition 15.11 and Remark 15.12).

Observe that $C_{D}(\operatorname{GIso}_{(V,\psi)})=\operatorname{GIso}_{(V,\psi)}^{{\text{id}}}$ and $C_{D}(\operatorname{Iso}_{(V,\psi)})=\operatorname{DL}_{{\sim}}^{\operatorname{id}}(V,\psi,D)$. It follows by Lemma 15.5, (14.3), and (14.12) that:

[TABLE]

Remark 16.1*.*

Assume that $G_{\mathcal{M}_{{\sim}}(M),1}^{0}$ is connected. By (2.8), Corollary 15.4, Proposition 15.6(a), and Remark 15.10 we observe that the equality

[TABLE]

is equivalent to the equality:

[TABLE]

Remark 16.2*.*

In [Se5, p. 380] there are examples of the computation of $\operatorname{MMT}_{{\sim}}(M)=G_{\mathcal{M}_{{\sim}}(M)}$. In [BK1, Theorems 7.3, 7.4], we computed $\operatorname{MMT}_{{\text{ahc}}}(M)$ for abelian varieties of dimension $\leq 3$ and families of abelian varieties of type I, II, and III in the Albert classification.

Remark 16.3*.*

If Serre’s Conjecture 15.9 holds for $M$, then by (15.2), (15.3), and Remark 15.10 we would have:

[TABLE]

In particular (16.5) and (16.6) hold for AHC motives in $\mathcal{M}_{{\text{ahc}}}$ (cf. Remark 15.12).

Following Serre [Se5, sec. 13, p. 396–397] we make the following definition.

Definition 16.4**.**

The algebraic Sato–Tate group $\operatorname{AST}_{K}(M)$ is defined as follows:

[TABLE]

Every maximal compact subgroup of $\operatorname{AST}_{K}(M)(\mathbb{C})$ will be called a Sato–Tate group associated with $M$ and denoted $\operatorname{ST}_{K}(M)$.

Remark 16.5*.*

In [BK2, Def. 11.7] we defined the algebraic Sato–Tate group for motives in $\mathcal{M}_{{\text{ahc}}}$ for which the Serre Conjecture 15.9 holds. In this paper we decided not to assume Conjecture 15.9 in Definition 16.7.

Remark 16.6*.*

Theorem 16.8 and Corollaries 16.11 and 16.12 below show that $\operatorname{AST}_{K}(M)$ is a natural candidate for the algebraic Sato–Tate group for the homogeneous motive $M$.

Corollary 16.7**.**

The following sequence is exact:

[TABLE]

Moreover:

[TABLE]

Proof.

The sequence in this corollary is just the exact sequence (15.4) with

[TABLE]

Hence (16.8) (resp. (16.9)) follows from Proposition 15.6(b) (resp. Proposition 15.6(c)). ∎

Theorem 16.8**.**

$\operatorname{AST}_{K}(M)$* is reductive and has the following properties:*

[TABLE]

Proof.

The group $\operatorname{AST}_{K}(M)$ is reductive by Theorem 15.2. Moreover (16.10) is (14.13) and (16.11) is just (15.10). Property (16.12) follows from Corollary 16.7. Because $\operatorname{AST}_{K}(M)^{\circ}(\mathbb{C})$ is a connected complex Lie group and any maximal compact subgroup of a connected complex Lie group is a connected real Lie group, the equality (16.13) follows. Property (16.14) follows by (15.3). ∎

Remark 16.9*.*

Recall (see Remark 15.10) that $\operatorname{AST}_{K}(M)^{\circ}=\operatorname{H}(V,\psi)$ is equivalent to Serre’s Conjecture 15.14.

Remark 16.10*.*

By Proposition 15.6(d) for the odd weight $n$:

[TABLE]

Now we discuss conditions under which the Algebraic Sato–Tate is amenable to computations and the Algebraic Sato–Tate conjecture holds.

Consider the following commutative diagrams with exact rows. Exactness of the rows of Diagram 16.2 follows by Corollary 15.3. The left and middle vertical arrows of Diagram 16.1 are monomorphisms by (16.10).

Corollary 16.11**.**

Assume that $\operatorname{H}(V,\psi)=C_{D}(\operatorname{Iso}_{(V,\psi)})$. Then

[TABLE]

Proof.

By the assumption and (16.2) we obtain:

[TABLE]

Hence (16.16) follows immediately from Diagram 16.2. Moreover, the middle vertical arrow in Diagram 16.2, which is the right vertical arrow in Diagram 16.1, is an isomorphism. Because $\operatorname{L}(V,\psi,D)=(C_{D}\operatorname{Iso}_{(V,\psi)})^{\circ}$, the assumption and the connectedness of $\operatorname{H}(V,\psi)$ yield $\operatorname{H}(V,\psi)=\operatorname{L}(V,\psi,D)$. Hence by (16.11), the left vertical arrow in Diagram 16.1 is an isomorphism, and so the middle vertical arrow in Diagram 16.1 is an isomorphism. ∎

Consider the following commutative diagrams with exact rows. Exactness of the rows of Diagram 16.4 follows by Corollary 5.2. The left and middle vertical arrows of Diagram 16.3 are monomorphisms by (4.22).

Corollary 16.12**.**

Assume that $\operatorname{H}(V,\psi)=C_{D}(\operatorname{Iso}_{(V,\psi)})$ and the Mumford–Tate conjecture holds for $M$. Then:

[TABLE]

Moreover, the algebraic Sato–Tate conjecture holds:

[TABLE]

Proof.

By (15.3) and Corollary 16.11:

[TABLE]

By Remark 7.4, the Mumford–Tate conjecture (7.3) holds if and only if the equality (7.6) holds, namely:

[TABLE]

Hence by the assumptions and connectedness of $\operatorname{H}(V,\psi)$, the left vertical arrow in Diagram 16.3 is an isomorphism and so (16.18) follows. Moreover by (4.16) and the assumption, we observe that $(G_{l,K,1}^{\operatorname{alg}})^{\circ}=(G_{l,K,1}^{\operatorname{alg}})^{\operatorname{id}}$. Hence

[TABLE]

Hence (16.19) follows from Diagram 16.4 and all vertical arrows in Diagrams 16.3 and 16.4 are isomorphisms. Applying also (16.21), the equality (16.20) follows. ∎

Remark 16.13*.*

V. Cantoral-Farfán and J. Commelin have proved [CF-C] that for abelian motives in $\mathcal{M}_{{\rm{mot}}}$, the Mumford-Tate conjecture implies the Algebraic Sato-Tate conjecture.

17. Computation of the identity connected component of $\operatorname{AST}_{K}(M)$

Consider the Tate motive $\mathbb{T}:=\mathbb{L}^{-1}=h^{2}(\mathbb{P}^{1})^{-1}=\mathbb{Q}(1)$ and the motivic Galois groups $G_{\mathcal{M}_{{\sim}}(M)}$, $G_{\mathcal{M}_{{\sim}}(M\oplus\mathbb{T})}$, and $G_{\mathcal{M}_{{\sim}}(\mathbb{T})}=\mathbb{G}_{m}$. By the definition of the motivic Galois group, there is a natural monomorphism:

[TABLE]

The projections

[TABLE]

are epimorphisms because categories $\mathcal{M}_{{\sim}}(M)$, $\mathcal{M}_{{\sim}}(M\oplus\mathbb{T})$ and $\mathcal{M}_{{\sim}}(\mathbb{T})$ are strictly full and fully faithful subcategories of the semisimple category $\mathcal{M}_{{\sim}}$. Hence they are also semisimple categories and the corresponding functors among them are also fully faithful (see [DM, Prop. 2.21, p. 139], cf. [Pan, p. 472]).

Definition 17.1**.**

Define the following groups:

[TABLE]

Recall that we denote $V=H_{B}(M)$. The same proof as the proof of (15.8) of Lemma 15.5 gives:

[TABLE]

By (2.7), (17.1), and the definitions of $\widetilde{\operatorname{MT}}(V,\psi)$ and $G_{{\mathcal{M}_{{\sim}}(M\oplus\mathbb{T})}}$, the following diagram with exact columns commutes, with (17.3) as the left arrow in the middle row:

From Diagram 17.1 we obtain natural isomorphism:

[TABLE]

Theorem 17.2**.**

The following natural map is an isomorphism:

[TABLE]

In addition there is the following equality:

[TABLE]

Proof.

Consider the following exact sequence:

[TABLE]

Because of Diagram 17.1 the cocharacter:

[TABLE]

splits $N$ in the exact sequence:

[TABLE]

Hence by Lemma 2.2,

[TABLE]

Observe that the commutative Diagram 15.2 holds for the motive $M\oplus\mathbb{T}$ so we obtain the isomorphism (17.5) from the right exact column of this diagram for $M\oplus\mathbb{T}$. ∎

Lemma 17.3**.**

There is the following commutative diagram with exact rows and columns. The map $\pi_{1}$ is induced by $\pi$.

Proof.

Observe that the right square of the Diagram 17.2 is the front face of the following cube Diagram 17.2.

Observe that the 5 remaining faces of Diagram 17.3 commute (the rear face commutes by Lemma 2.2). Hence the front face also commutes by (17.4) and a diagram chase. ∎

By the definition of $N$ and (17.1), the kernel of $\pi$ in Diagram 17.2 is contained in ${\rm{Id}}_{V}\times\mathbb{G}_{m}$. Hence the restriction of $N$ to the kernel of $\pi$ is a monomorphism. By the commutativity of Diagram 17.2, the kernel of $\pi$ injects into $\mu_{n}$. Hence $\pi_{1}$ is a monomorphism. In addition $\dim G_{\mathcal{M}_{{\sim}}(M\oplus\mathbb{T})}=\dim G_{\mathcal{M}_{{\sim}}(M)}$ and consequently $\dim G_{\mathcal{M}_{{\sim}}(M\oplus\mathbb{T}),1}=\dim G_{\mathcal{M}_{{\sim}}(M),1}$. This shows that

[TABLE]

Consider the following commutative diagram.

The top row is the isomorphism of Theorem 17.2. The bottom row is the exact sequence of Theorem 15.2. The map $\overline{\pi}$ in Diagram 17.4 is an epimorphism because the map $\pi$ in Diagram 17.2 is an epimorphism. The map $\overline{\pi}_{1}$ in Diagram 17.4 is a monomorphism because the map $\pi_{1}$ in Diagram 17.2 is a monomorphism and because of (17.9). Proposition 15.6(b) and a chase in Diagram 17.4 show that

[TABLE]

where $-{\rm{Id}}_{V}(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}$ denotes the coset of $-{\rm{Id}}_{V}$ in the quotient group $\pi_{0}(G_{\mathcal{M}_{{\sim}}(M),1})=G_{\mathcal{M}_{{\sim}}(M),1}/(G_{\mathcal{M}_{{\sim}}(M),1})^{\circ}$.

Consider the following commutative diagram with exact rows.

The equality (17.10) and a chase in Diagram 17.5 show that

[TABLE]

In particular $[G_{\mathcal{M}_{{\sim}}(M),1}:\,G_{\mathcal{M}_{{\sim}}(M\oplus\mathbb{T}),1}]\leq 2$ and $G_{\mathcal{M}_{{\sim}}(M\oplus\mathbb{T}),1}\,\triangleleft\,G_{\mathcal{M}_{{\sim}}(M),1}$.

To be consistent with Serre’s approach, as in §6 and 7, we make the following definition.

Definition 17.4**.**

The algebraic Sato–Tate group $\widetilde{\operatorname{AST}_{K}}(M)$ is defined as follows:

[TABLE]

Every maximal compact subgroup of $\widetilde{\operatorname{AST}_{K}}(M)(\mathbb{C})$ will be called a Sato–Tate group associated with $M$ and denoted $\widetilde{\operatorname{ST}_{K}}(M)$.

Corollary 17.5**.**

The following equalities hold:

• (a)

$\operatorname{AST}_{K}(M)^{\circ}=G^{0}_{\mathcal{M}_{{\sim}}(M\oplus\mathbb{T}),1}$.

• (b)

$\operatorname{AST}_{K}(M)=\widetilde{\operatorname{AST}_{K}}(M)\,\cup\,-{\rm{Id}}_{V}\widetilde{\operatorname{AST}_{K}}(M)$.

• (c)

$\operatorname{ST}_{K}(M)=\widetilde{\operatorname{ST}_{K}}(M)\,\cup\,-{\rm{Id}}_{V}\widetilde{\operatorname{ST}_{K}}(M)$* up to conjugation in *

$\operatorname{AST}_{K}(M)(\mathbb{C})$.

Proof.

(a) follows by (17.6), (b) follows by (17.11), and (c) follows by (b). ∎

Consider the following commutative Diagram 17.6 of group schemes with exact columns. The middle horizontal arrow is a closed immersion by Remark 15.2. This remark obviously holds for all motives in $\mathcal{M}_{{\sim}}$, not only polarized motives. The middle horizontal arrow is a closed immersion. Hence the top horizontal arrow is also a closed immersion.

By Definition 17.12 and Diagram 17.6, we obtain:

[TABLE]

Hence Algebraic Sato–Tate Conjecture 7.19(a) holds for $\widetilde{\operatorname{AST}_{K}}(M)$ with respect to the representation

[TABLE]

Algebraic Sato–Tate Conjecture 7.19(b) for $\widetilde{\operatorname{AST}_{K}}(M)$ states:

[TABLE]

Proposition 17.6**.**

The Algebraic Sato–Tate conjecture holds for $\operatorname{AST}_{K}(M)$ (Conjecture 7.5) if and only if the Algebraic Sato–Tate conjecture holds for $\widetilde{\operatorname{AST}_{K}}(M)$ (Conjecture 7.19).

Proof.

This follows by (16.14), (17.13), (6.7) and Corollary 17.5(b). ∎

Because of Proposition 17.6, it is enough to discuss in the following sections the Algebraic Sato–Tate conjecture only for $\operatorname{AST}_{K}(M)$.

Proposition 17.7**.**

The Sato–Tate conjecture for $\operatorname{ST}_{K}(M)$ (Conjecture 7.12) implies the Sato–Tate conjecture for $\widetilde{\operatorname{ST}_{K}}(M)$ (Conjecture 7.33).

Proof.

This follows by Theorem 7.43. ∎

Recall that $D:={{\rm End}}_{\mathcal{M}_{\overline{K},\sim}}(\overline{M})$ (see (13.8)).

Proposition 17.8**.**

Put $\widetilde{\mathcal{G}_{K}^{\operatorname{alg}}}:=G_{\mathcal{M}_{{\sim}}(M\oplus\mathbb{T})}$ and $\mathcal{G}_{K}^{\operatorname{alg}}:=G_{\mathcal{M}_{{\sim}}(M)}$. Assume that the family $(V_{l})_{l}$ of $l$-adic realizations of $M$ is a strictly compatible family of $l$-adic representations $(\rho_{l})_{l}$ in the sense of Serre. Then Conjectures 8.1(a) and 8.2(a) hold for the family $(\rho_{l})_{l}$.

Proof.

By Lemmas 13.9, 13.10, 13.12, and the assumption of strict compatibility, the conditions (D1), (D2), (DR1), (DR2), (R1)–(R4) of §2, §3, and §4 hold for the Betti, de Rham and $l$-adic realizations of $M$. Observe that the middle left horizontal arrow in Diagram 15.1 and the middle horizontal arrow in Diagram 17.6 are closed immersions. Hence the Proposition follows by the properties of the morphisms (17.1) and (17.2). ∎

Corollary 17.9**.**

Under the notation and assumptions of Proposition 17.8, Conjectures 8.1, 8.2, 7.5, and 7.19 are equivalent for the family of $l$-adic representations $(\rho_{l})_{l}$.

Proof.

This follows from Proposition 17.8 and Corollary 8.18. ∎

Remark 17.10*.*

Recall the category of polarized realizations $R_{K}^{\rm{p}}$ from §9. Deligne’s category $\mathcal{M}_{{\text{ahc}}}$ of motives for absolute Hodge cycles is a full subcategory of $R_{K}^{\rm{p}}$ via the Betti, de Rham and $l$-adic realizations.

Remark 17.11*.*

It is not known in general that the family of $l$-adic representations coming from $l$-adic realizations of a motive in $\mathcal{M}_{{\text{ahc}}}$ or $\mathcal{M}_{{\text{mot}}}$ is strictly compatible in the sense of Serre.

We would like to propose the following Geometricity Conjecture.

Conjecture 17.12**.**

Objects of $R_{K}^{\rm{p}}$ are realizations of motives from $\mathcal{M}_{{\text{ahc}}}$. More precisely, let $\mathcal{V}\,:=\,(\,V_{{}_{\rm{DR}}},\,(V_{l})_{l},\,(V_{\sigma})_{\sigma},\,I_{\infty,\sigma},\,I_{l,\bar{\sigma}})\in{\rm{obj}}(R_{K}^{\rm{p}})$ be a pure, polarized realization with components satisfying conditions (D1), (D2), (DR1), (DR2), (R1)–(R4) of §2, §3, and §4 respectively with $D$ as in Definition 9.12. Then $\mathcal{V}$ comes from a pure motive of the Deligne motivic category for absolute Hodge cycles $\mathcal{M}_{K,{{\rm{{{\rm{ahc}}}}}}}$ via the Betti, de Rham and $l$-adic realizations.

Remark 17.13*.*

Conjecture 17.12 might be formulated for other motivic categories. Nevertheless the Betti, de Rham and $l$-adic realizations functor $\mathcal{M}_{K,{{}_{\sim}}}(M)\rightarrow R_{K}^{\rm{p}}$ factors through $\mathcal{M}_{K,{{\rm{{{\rm{ahc}}}}}}}$. Hence the Geometricity Conjecture is stated in the weakest possible form.

Recall that there is also a natural epimorphism (see Remark 7.42):

[TABLE]

Definition 17.14**.**

The Sato–Tate parity group for a motive $M$ is defined as follows (cf. Corollary 17.5):

[TABLE]

Remark 17.15*.*

While $\operatorname{P}_{\rm{ST}}(V,\psi)$ (see (8.19)) is defined assuming Conjectures 8.1(a) and 8.2(a) for the family of $l$-adic representations $(\rho_{l})_{l}$, the group $\operatorname{P}_{\rm{ST}}(M,\psi_{{\sim}})$ is defined unconditionally for motives $M$ for which $l$-adic realizations give a family of $l$-adic representations $(\rho_{l})_{l}$ which is strictly compatible in the sense of Serre (see Lemmas 13.9, 13.10, and 13.12 and Proposition 17.8).

Now Theorem 8.21 has a counterpart for motives.

Theorem 17.16**.**

Assume that $l$-adic realizations of the motive $M$ give a family of $l$-adic representations $(\rho_{l})_{l}$ which is strictly compatible in the sense of Serre.

• (a)

If $\operatorname{P}_{\rm{ST}}(M,\psi_{{\sim}})$ is nontrivial, i.e. $-{{\rm Id}}_{V}\notin\widetilde{\operatorname{ST}_{K}}(M)$, then Sato–Tate Conjecture 7.12 does not hold.

• (b)

If $\operatorname{P}_{\rm{ST}}(M,\psi_{{\sim}})$ is trivial, i.e. $-{{\rm Id}}_{V}\in\widetilde{\operatorname{ST}_{K}}(M)$, then Sato–Tate Conjectures 7.12 and 7.33 are equivalent.

Proof.

The proof is the same as for Theorem 7.43. ∎

Remark 17.17*.*

Theorem 17.16(a) can be illustrated using Example 2.16, as the Sato–Tate conjecture is known for $A$ when $K$ is a totally real field [HSBT], [BLGG] or a CM field [ACC*+*]. In these cases, Conjecture 7.33 holds but Conjecture 7.12 does not.

By Definitions 15.21, 17.14, and equality (17.9) there is a natural homomorphism:

[TABLE]

The groups $\operatorname{P}_{S}(M,\psi_{{\sim}})$ and $\operatorname{P}_{\rm{ST}}(M,\psi_{{\sim}})$ have order at most 2. These groups are generated by the corresponding cosets of $-{\rm{Id}}_{V}$. Moreover the homomorphism (17.14) carries the coset of $-{\rm{Id}}_{V}$ in the source into the coset of $-{\rm{Id}}_{V}$ in the target. Hence the homomorphism (17.14) is an epimorphism, cf. Remark 15.21.

Consider the following commutative diagram.

The map $\pi$ is an epimorphism cf. (17.2). Consequently $\overline{\pi}$ is an epimorphism. By Diagram 17.2 of Lemma 17.2 and the comments below the proof of this lemma, the map $\pi$ has finite kernel. Hence the algebraic groups $G_{\mathcal{M}_{{\sim}}(M\oplus\mathbb{T})}$ and $G_{\mathcal{M}_{{\sim}}(M)}$ have the same dimension and the map $\pi^{\circ}$ has finite kernel. Because the image of $\pi^{\circ}$ is a closed subgroup of $(G_{\mathcal{M}_{{\sim}}(M)})^{\circ}$ [Hu, Section 7.4, Prop. B(b)] of the same dimension as $(G_{\mathcal{M}_{{\sim}}(M\oplus\mathbb{T})})^{\circ}$, the image is also open. Because the group $(G_{\mathcal{M}_{{\sim}}(M)})^{\circ}$ is connected, the map $\pi^{\circ}$ must be an epimorphism. By the snake lemma applied to Diagram 17.7, there is the following short exact sequence:

[TABLE]

Proposition 17.18**.**

There is the following short exact sequence:

[TABLE]

Proof.

By Diagram 17.5 and Definitions 15.8, 16.7, and 17.12 we have:

[TABLE]

Now the claim follows by the snake lemma applied to Diagram 17.4. ∎

18. Equidistribution of Frobenii in $l$-adic realization of motives

As before, let $M$ be a direct factor of $h_{{\sim}}^{r}(X)(m)$. By definition, $M$ is a homogeneous motive of weight $n=2m-r$. Recall that $H^{r}_{l}$ denotes the $l$-adic realization of $h_{{\sim}}^{r}(X)(m)$ (see §13). Let $\rho_{l}$ be the $l$-adic representation coming from the $l$-adic realizations of $M$. Consider the finite Galois extension $K_{0}/K$ associated with $\rho_{l}$ according to (5.14) and (5.15).

Theorem 18.1**.**

Assume that the family $(\rho_{l})$ is strictly compatible. Assume that $n\not=0$ and there is $c\in\mathbb{N}$ such that $(\mathbb{Z}_{l}^{\times})^{c}\,\,{\rm{Id}}_{H^{r}_{l}}\,\subset\,\rho_{H^{r}_{l}}(G_{K})$ for all $l$. Moreover assume that for some $l$ coprime to $c$:

• (1)

$K_{0}\,\cap\,K(\mu_{\bar{l}}^{\otimes\,n})\,=\,K$,

• (2)

$\operatorname{ast}_{l,K}$* is an isomorphism with respect to $\rho_{l}$.*

Then the Sato–Tate Conjecture holds for the representation $\rho_{l}\colon G_{K}\rightarrow\operatorname{GIso}_{(V_{l},\psi_{l})}(\mathbb{Q}_{l})$ (resp. $\widetilde{\rho}_{l}\colon G_{K}\rightarrow\operatorname{GL}(V_{l}\oplus\mathbb{Q}_{l}(1))$) with respect to $\operatorname{ST}_{K}(M)$ (resp. $\widetilde{\operatorname{ST}_{K}}(M)$) if and only if it holds for $\rho_{l}\colon G_{K_{1}}\rightarrow\operatorname{GIso}_{(V_{l},\psi_{l})}(\mathbb{Q}_{l})$ (resp. $\widetilde{\rho}_{l}\colon G_{K_{1}}\rightarrow\operatorname{GL}(V_{l}\oplus\mathbb{Q}_{l}(1))$) with respect to $\operatorname{ST}_{K_{1}}(M)$ (resp. $\widetilde{\operatorname{ST}_{K_{1}}}(M)$) for all subextensions $K_{1}$ of $K_{0}/K$ such that $K_{0}/K_{1}$ is cyclic (cf. Theorem 11.12).

Proof.

Lemmas 13.9, 13.10 and 13.12 show that $\rho_{l}$ satisfies conditions (D1), (D2), (DR1), (DR2) and (R1)–(R4). The category $\mathcal{M}_{{\rm{\sim}}}$ is abelian semisimple by assumption, hence $V_{l}$ is a direct factor of $H_{l}$ as a $\mathbb{Q}_{l}[G_{K}]$-module. Hence $(\mathbb{Z}_{l}^{\times})^{c}\,\,{\rm{Id}}_{V_{l}}\,\subset\,\rho_{l}(G_{K})$ for all $l$. Since $l$ is coprime to $c$ then $1+l\,\mathbb{Z}_{l}\subset(\mathbb{Z}_{l}^{\times})^{c}$. Therefore we can apply directly Theorem 11.12. ∎

Remark 18.2*.*

Let $A/K$ be an abelian variety with $\dim A\leq 4$. Then by [W, p. 5 Corollary 1] the Lang conjecture holds, i.e. $\rho_{l}(G_{K})$ contains the group of all homotheties in $\operatorname{GL}_{T_{l}(A)}(\mathbb{Z}_{l})$ for all $l\gg 0$.

Remark 18.3*.*

If $n$ is odd, then by Propositions 5.14 and 7.16 the field $K_{0}$ is independent of $l$. Hence for $l$ big enough $K_{0}\,\cap\,K(\mu_{\bar{l}}^{\otimes\,n})\,=\,K$.

Remark 18.4*.*

Let $A/K$ be an abelian variety with $\dim A\leq 3$. Then by [BK1, Theorem 6.11] the homomorphism $\operatorname{ast}_{l,K}$ is an isomorphism with respect to $\rho_{l}$.

Remark 18.5*.*

Assume that Algebraic Sato–Tate Conjecture 7.5 holds for $M$. By Proposition 7.9, $\operatorname{ST}_{K_{0}}(M)$ is connected if and only if $\operatorname{P}(V_{l},\psi_{l})$ is trivial (cf. Definition 5.18). In particular $\operatorname{ST}_{K_{0}}(M)$ is connected when $n$ is odd (see Theorem 5.6). Hence in some cases Theorems 10.20 and 18.1 provide an opportunity to check the Sato–Tate conjecture on the identity connected component of the Sato–Tate group.

Example 18.6**.**

Let $A/K$ be an abelian surface over a number field. Consider the motive $h_{{\rm{ahc}}}^{1}(A)$. Put $\operatorname{ST}_{K}(A):=\operatorname{ST}_{K}(h_{{\rm{ahc}}}^{1}(A))$. Observe that $\operatorname{ST}_{K}(A)^{\circ}=\operatorname{ST}_{K_{0}}(A)$ by Remark 18.5. It was shown in [FKRS, Table 8, p. 1434] that there are 52 possibilities for $\operatorname{ST}_{K}(A)$ and each quotient $\operatorname{ST}_{K}(A)/\operatorname{ST}_{K_{0}}(A)$ is solvable. Taking into account Remarks 18.2, 18.3, 18.4 we observe that all assumptions of Theorem 18.1 hold in this case. The identity connected component $\operatorname{ST}_{K_{0}}(A)$ in every of these 52 cases is isomorphic to one of the following 6 groups: $\operatorname{U}(1)$, $\operatorname{SU}(2)$, $\operatorname{U}(1)\times\operatorname{U}(1)$, $\operatorname{U}(1)\times\operatorname{SU}(2)$, $\operatorname{SU}(2)\times\operatorname{SU}(2)$, $\operatorname{USp}(4)$ (see [FKRS, Table 1, p. 1402]).

Remark 18.7*.*

The Sato–Tate conjecture for abelian surfaces was proven in many cases by Christian Johansson [Jo]. In the cases where $\operatorname{ST}_{K_{0}}(A)$ is one of $\operatorname{U}(1)$, $\operatorname{U}(1)\times\operatorname{U}(1)$, $\operatorname{U}(1)\times\operatorname{SU}(2)$, $\operatorname{USp}(4)$, by Theorem 10.20 it is only necessary to check the Sato–Tate conjecture for $\operatorname{ST}_{K_{0}}(A)$ itself. In the case where $\operatorname{ST}_{K_{0}}(A)=\operatorname{SU}(2)\times\operatorname{SU}(2)$, one must separately treat the case where $\operatorname{ST}_{K}(A)=N(\operatorname{SU}(2)\times\operatorname{SU}(2))$. In the case where $\operatorname{ST}_{K_{0}}(A)=\operatorname{SU}(2)$, one must separately treat several additional cases.

Remark 18.8*.*

Let $A/K$ be an abelian variety such that over some finite extension of $K$, $A$ becomes isogenous to a product of abelian varieties with complex multiplication. Then the Sato–Tate conjecture is known for $A$; see for example [Jo, Proposition 16]. This includes some examples where $\operatorname{ST}_{K}(A)/\operatorname{ST}_{K_{0}}(A)$ is not solvable; for example, in [FKS] one finds an example of an abelian threefold for which $\operatorname{ST}_{K}(A)/\operatorname{ST}_{K_{0}}(A)$ is simple of order 168. One can make higher-dimensional examples using the method of [GK].