On the intersection motive of certain Shimura varieties: the case of Siegel threefolds

TL;DR

This paper constructs a Hecke-equivariant Chow motive for Siegel threefolds, linking intersection cohomology to motives and enabling the definition of Grothendieck motives for Siegel modular forms.

Contribution

It introduces a new construction of a Chow motive for Siegel threefolds that captures intersection cohomology and facilitates the study of modular forms through motives.

Findings

Construction of a Hecke-equivariant Chow motive for Siegel threefolds

Realizations of the motive match intersection cohomology with algebraic coefficients

Enables the definition of Grothendieck motives for Siegel modular forms

Abstract

In this article, we construct a Hecke-equivariant Chow motive whose realizations equal intersection cohomology of Siegel threefolds with regular algebraic coefficients. As a consequence, we are able to define Grothendieck motives for Siegel modular forms.

Equations

$$i_{\ast }{i}^{\ast }{j}_{\ast }\text{mathds}{1}_M[-1]⟶j_{!}\text{mathds}{1}_M⟶j_{\ast }\text{mathds}{1}_M⟶i_{\ast }{i}^{\ast }{j}_{\ast }\text{mathds}{1}_M$$

$$G:=GSp(V,J)\subset GL(V).$$

$$\{g\in \mathrm{GL}(V{\otimes }_{\mathbb{Q}}R),\exists \lambda (g)\in R^{\ast },J(g\bullet ,g\bullet )=\lambda (g)\cdot J(\bullet ,\bullet )\}.$$

$$\lambda :G⟶{\mathbb{G}}_{m,\mathbb{Q}}.$$

$$\left(\begin{array}[]{cc}0&I_{2}\\ -I_{2}&0\end{array}\right)\;,$$

$${}^{t}gJg=\lambda (g)\cdot J,$$

$$\{\mathrm{diag}(a,b,a^{-1}q,b^{-1}q)\in G{L}_{4,\mathbb{Q}}\},$$

$$r\equiv k_1+k_2\mathrm{mod}2.$$

$$\alpha (k_1,k_2,r):T⟶{\mathbb{G}}_{m,\mathbb{Q}},\mathrm{diag}(a,b,a^{-1}q,b^{-1}q)⟼a^{k_1}{b}^{k_2}{q}^{-\frac{r+k_1+k_2}{2}}.$$

$${\mathbb{G}}_{m,\mathbb{Q}}⟶T,x⟼\mathrm{diag}(x,x,x,x)$$

$${\mathbb{G}}_{m,\mathbb{Q}}⟶{\mathbb{G}}_{m,\mathbb{Q}},x\mapsto x^{-r}.$$

$$\mathcal{H}:=\{\tau \in M_2(\mathbb{C}),{}^t\tau =\tau \text{and}Im(\tau )\text{definite}\}.$$

$$r\equiv k_1+k_2\mathrm{mod}2,$$

$$k_1\ge k_2\ge 0.$$

$$\widetilde{\mu}:\mathop{\rm Rep}\nolimits(G)\longrightarrow CHM^{s}\bigl{(}M^{K}\bigr{)}_{\mathbb{Q}}$$

$${}^{\underline{\alpha }}\mathcal{V}:=\tilde{\mu }(V_{\underline{\alpha }}).$$

$$-k+1,-k+2,\dots ,k,$$

$$a_{\ast }{j}_{!\ast }{}^{\underline{\alpha }}\mathcal{V}\in CHM(\mathbb{Q}{)}_{\mathbb{Q}},$$

$$a_{\ast }{i}_{\ast }{i}^{\ast }{j}_{!\ast }{}^{\underline{\alpha }}\mathcal{V}[-1]⟶{\tilde{a}}_{!}{}^{\underline{\alpha }}\mathcal{V}⟶a_{\ast }{j}_{!\ast }{}^{\underline{\alpha }}\mathcal{V}⟶a_{\ast }{i}_{\ast }{i}^{\ast }{j}_{!\ast }{}^{\underline{\alpha }}\mathcal{V}$$

$$a_{\ast }{j}_{!\ast }{}^{\underline{\alpha }}\mathcal{V}⟶{\tilde{a}}_{\ast }{}^{\underline{\alpha }}\mathcal{V}⟶a_{\ast }{i}_{\ast }{i}^{!}{j}_{!\ast }{}^{\underline{\alpha }}\mathcal{V}[1]⟶a_{\ast }{j}_{!\ast }{}^{\underline{\alpha }}\mathcal{V}[1]$$

$${}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}\lhook\joinrel\longrightarrow\pi_{N,*}\mathds{1}_{{\rm{B}}^{N}}(\ell)[2\ell]\mathrel{\vbox{\offinterlineskip\hbox spread13.99995pt{\hfil$\scriptstyle\ $\hfil}\hbox to24.16661pt{\rightarrowfill\hskip-7.96677pt$\rightarrow$}}}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}\;,$$

$$a_{\ast }m:{\tilde{a}}_{!}{\pi }_{N,\ast }\text{mathds}{1}_{{\mathrm{B}}^N}⟶{\tilde{a}}_{\ast }{\pi }_{N,\ast }\text{mathds}{1}_{{\mathrm{B}}^N}$$

$$a_{\ast }m:{\tilde{a}}_{!}{}^{\underline{\alpha }}\mathcal{V}⟶{\tilde{a}}_{\ast }{}^{\underline{\alpha }}\mathcal{V}.$$

$$H^{n}\bigl{(}(M^{K})^{*}({\mathbb{C}}),\mathop{j_{!*}}\nolimits\mu_{\bf H}(V_{\mathop{\underline{\alpha}}\nolimits})\bigr{)}\longrightarrow H^{n}\bigl{(}M^{K}({\mathbb{C}}),\mu_{\bf H}(V_{\mathop{\underline{\alpha}}\nolimits})\bigr{)}$$

$$H^{n}\bigl{(}(M^{K})^{*}\times_{\mathbb{Q}}\bar{{\mathbb{Q}}},\mathop{j_{!*}}\nolimits\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})\bigr{)}\longrightarrow H^{n}\bigl{(}M^{K}\times_{\mathbb{Q}}\bar{{\mathbb{Q}}},\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})\bigr{)}$$

$$H^{n}_{c}\bigl{(}M^{K}({\mathbb{C}}),\mu_{\bf H}(V_{\mathop{\underline{\alpha}}\nolimits})\bigr{)}\longrightarrow H^{n}\bigl{(}(M^{K})^{*}({\mathbb{C}}),\mathop{j_{!*}}\nolimits\mu_{\bf H}(V_{\mathop{\underline{\alpha}}\nolimits})\bigr{)}$$

$$H^{n}_{c}\bigl{(}M^{K}\times_{\mathbb{Q}}\bar{{\mathbb{Q}}},\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})\bigr{)}\longrightarrow H^{n}\bigl{(}(M^{K})^{*}\times_{\mathbb{Q}}\bar{{\mathbb{Q}}},\mathop{j_{!*}}\nolimits\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})\bigr{)}$$

$$\bigl{(}{\mathfrak{e}}-{\mathfrak{e}}^{2}\bigr{)}^{N}=0\;.$$

$$e_{{\mathbb{Z}}_{p}}:=\bigl{(}{\rm id}_{\widetilde{{\cal B}^{N}}}-({\rm id}_{\widetilde{{\cal B}^{N}}}-{\mathfrak{e}})^{N}\bigr{)}^{N}$$

$$\partial (M^K{)}^{\ast }=Z_0\coprod Z_1;$$

Full text

Keywords: Siegel threefolds, weight structures, intersection motive, motives for Siegel modular forms.

Math. Subj. Class. (2010) numbers: 14G35 (11F32, 11F46, 14C25, 14F20, 14F25, 14F30, 14F32, 19E15, 19F27).

1. 0 Introduction
2. 1 Statement of the main result
3. 2 Proof of the main result
4. 3 The motive for an automorphic form

0 Introduction

The purpose of this paper is the construction and analysis of the intersection motive of Kuga–Sato families over a Siegel threefold relative to its Satake–(Baily–Borel) compactification. As in earlier work on Hilbert–Blumenthal varieties [Wi5], Picard surfaces [Wi6], and more generally, Picard varieties of arbitrary dimension [C], the use of the formalism of weight structures [Bo] proves to be successful for dealing with a problem, for which explicit geometrical methods seem inefficient.

However, Siegel threefolds present a characteristic feature different from the cases treated so far: the dimension of the boundary of their Satake–(Baily–Borel) compactification is equal to one. In particular, it is strictly positive.

As a consequence, the context of geometrical motives, i.e., motives over a point, is no longer adapted to the problem. Let us explain why.

The present construction, as the preceding ones, depends on absence of weights $-1$ and [math] in the boundary motive. To prove absence of weights, the idea remains, as previously, to employ realizations. But then, realizations need to detect weights (and therefore, their absence). One may expect this to be true in general; let us agree to refer to that principle as weight conservativity. To date, weight conservativity is proved for the restriction of the (generic) $\ell$-adic realization to the category of motives of Abelian type of characteristic zero [Wi8].

However, unless the boundary of the Baily–Borel compactification of a given Shimura variety $M$ is of dimension zero, its boundary motive, as well as the boundary motive of any Kuga–Sato family ${\rm{B}}$ over $M$, is in general not of Abelian type; this is in any case true if $M$ is a Siegel threefold. Concretely, this means that even if the realization of the boundary motive were proved to avoid weights $-1$ and [math], we could not formally conclude that the same is true for the boundary motive itself.

This is where relative motives, together with the formalism of six operations enter. Denoting by $j$ the open immersion of $M$ into its Baily–Borel compactification $M^{*}$, by $i$ its closed complement, and by $\mathds{1}_{M}$ the structural motive over $M$, there is an exact triangle

[TABLE]

of motives over $M^{*}$. The boundary motive of $M$ is isomorphic to the dual of the direct image of $i_{*}i^{*}j_{*}\mathds{1}_{M}$ under the structure morphism of $M^{*}$. More generally, the boundary motive of ${\rm{B}}$ is isomorphic to the dual of the direct image of $i_{*}i^{*}j_{*}\pi_{*}(\mathds{1}_{\rm{B}})$, where $\pi:{\rm{B}}\to M$ denotes the projection of the Kuga–Sato family ${\rm{B}}$ to its base.

It is then true that the relative motive $i_{*}i^{*}j_{*}\pi_{*}(\mathds{1}_{\rm{B}})$ over $M^{*}$ is of Abelian type.

Whence our strategy of proof. First, identify the $\ell$-adic realization of $i_{*}i^{*}j_{*}\pi_{*}(\mathds{1}_{\rm{B}})$, or more generally, of $i_{*}i^{*}j_{*}{\cal V}$, for direct factors ${\cal V}$ of $\pi_{*}(\mathds{1}_{\rm{B}})$; in the cases where weights [math] and $1$ are avoided, weight conservativity tells us that $i_{*}i^{*}j_{*}{\cal V}$ itself avoids weights [math] and $1$. Second, apply the direct image $a_{*}$ associated to the structure morphism $a$ of $M^{*}$. It is proper, therefore, the functor $a_{*}$ is weight exact. In particular, if $i_{*}i^{*}j_{*}{\cal V}$ avoids weights [math] and $1$, then so does $a_{*}i_{*}i^{*}j_{*}{\cal V}$. The corresponding direct factor of the boundary motive of ${\rm{B}}$ thus avoids weights $-1$ and [math].

It may be useful to remark that if $M$ is a Hilbert–Blumenthal or Picard variety, then there is essentially no difference between $i_{*}i^{*}j_{*}{\cal V}$ and its direct image under $a$, since the latter is of relative dimension zero on the boundary of $M^{*}$.

The passage from geometrical motives to relative motives necessitates a certain number of technical adjustments. For better legibility, we decided to separate these from the present text. The result is [Wi9]; it contains in particular the identification of the boundary motive and the dual of $a_{*}i_{*}i^{*}j_{*}\pi_{*}(\mathds{1}_{\rm{B}})$ mentioned above.

Compared to the cases treated earlier, another feature of the boundary of Siegel threefolds is new: its canonical stratification is not reduced to a single type of strata. Indeed, in the boundary, one finds a closed stratum of dimension zero, the so-called Siegel stratum, and its complement, the so-called Klingen stratum, which is a disjoint union of (open) modular curves. Control of the weights avoided by the restrictions of the $\ell$-adic realization $R_{\ell}(i^{*}j_{*}\pi_{*}(\mathds{1}_{\rm{B}}))$ of $i^{*}j_{*}\pi_{*}(\mathds{1}_{\rm{B}})$ to the two strata is related to, but does not a priori determine the weights avoided by $R_{\ell}(i^{*}j_{*}\pi_{*}(\mathds{1}_{\rm{B}}))$. In fact, the precise relation is given by a long exact localization sequence. Its control it not obvious. In an earlier attempt, we succeeded to identify sufficiently many terms in this sequence, and (above all) certain morphisms, to prove absence of weights [math] and $1$. This approach is technically difficult; moreover, it does not use the auto-duality property of the coefficients. Indeed, the device dual to the localization sequence is the co-localization sequence; even when the coefficients are auto-dual, the two sequences cannot be related. It turns out that both problems admit the same solution. Namely, the theory of intermediate extensions allows to represent $R_{\ell}(i^{*}j_{*}\pi_{*}(\mathds{1}_{\rm{B}}))$ as an extension of two “halfs”, one of which dual to the other, and both related to the intermediate extension $\mathop{j_{!*}}\nolimits\pi_{*}(\mathds{1}_{\rm{B}})$. This observation is equally integrated in [Wi9]; for our purposes, its concrete interest is to divide by two the number of cohomological degrees for which absence of weights has to be tested, and to reduce the number of morphism in the localization sequence, which need to be identified, to zero.

The rôle of the intermediate extension is not only technical. It turns out that the dual of its direct image under $a$ is canonically isomorphic to the interior motive, which according to [Wi3] can be defined as soon as the boundary motive avoids weights $-1$ and [math]. This motivates the slight change of terminology in the title, as compared to the earlier work mentioned above [Wi5, Wi6, C].

Let us now give a more detailed account of the content of the present article. Section 1 contains the statement of our main result, Theorem 1.6. Denote by $GSp_{4,{\mathbb{Q}}}$ the group of symplectic similitudes of a fixed four-dimensional ${\mathbb{Q}}$-vector space $V$. As will be recalled, irreducible representations of $GSp_{4,{\mathbb{Q}}}$ are indexed by weights $\mathop{\underline{\alpha}}\nolimits$ depending on three integral parameters: $\mathop{\underline{\alpha}}\nolimits=\alpha(k_{1},k_{2},r)$. The weight $\mathop{\underline{\alpha}}\nolimits$ is dominant if and only if $k_{1}\geq k_{2}\geq 0$; it is regular if and only if $k_{1}>k_{2}>0$. Denote by $V_{\mathop{\underline{\alpha}}\nolimits}$ the irreducible representation of highest weight $\mathop{\underline{\alpha}}\nolimits$. According to the main result from [A] (which will be recalled in Theorem 1.4), there is a Chow motive ${}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ over the Siegel threefold $M$, whose cohomological (Hodge theoretic or $\ell$-adic) realizations equal the classical canonical construction $\mu(V_{\mathop{\underline{\alpha}}\nolimits})$. Part (a) of Theorem 1.6 then states that $i^{*}j_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ is of Abelian type. Part (b) asserts that if $\mathop{\underline{\alpha}}\nolimits$ is regular, then $i^{*}j_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ avoids weights [math] and $1$. It has recently become increasingly important to determine the precise interval containing $[0,1]$ of weights avoided by $i^{*}j_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$. Theorem 1.6 (b) gives a complete answer: putting $k:=\min(k_{1}-k_{2},k_{2})$, the motive $i^{*}j_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ avoids all the weights between $-k+1$ and $k$, while both weights $-k$ and $k+1$ do occur. Interestingly, this result does not depend on the level of the Siegel threefold. We then list the main consequences of this result (Corollaries 1.7, 1.8, 1.9, 1.11, 1.13), applying the theory developed in [Wi9].

Section 2 is devoted to the proof of Theorem 1.6. As in previous cases, our control of smooth toroidal compactifications of $M$ is sufficiently explicit to verify that, as stated in Theorem 1.6 (a), the motive $i^{*}j_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ is indeed of Abelian type. Given this result, and weight conservativity of the restriction of the $\ell$-adic realization $R_{\ell}$, part (b) of Theorem 1.6 may be checked on the image of $i^{*}j_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ under $R_{\ell}$. Given that ${}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ realizes to give $\mu(V_{\mathop{\underline{\alpha}}\nolimits})$, the restriction of $R_{\ell}(i^{*}j_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V})$ to the (Siegel and Klingen) strata can be computed following a standard pattern, employing Pink’s and Kostant’s Theorems. This computation (Theorem 2.3) is considerably simplified by results of Lemma’s [Lm]. It remains to glue the information coming from the strata, in order to get control of the weights on the whole boundary. The part of Theorem 1.6 (b) asserting that weights $-k$ and $k+1$ occur in $R_{\ell}(i^{*}j_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V})$ (Proposition 2.9) is the single ingredient requiring a proof longer than any other.

In the final Section 3, we give the necessary ingredients to perform the construction of the Grothendieck motive associated to a (Siegel) automorphic form with coefficients in an irreducible representation with regular highest weight (Definition 3.5). This is the analogue for Siegel threefolds of the main result from [Sc]. On the level of Galois representations, our definition coincides with Weissauer’s [We, Thm. I]. We also recover Urban’s result [U, Thm. 1] on characteristic polynomials associated to Frobenii (Corollary 3.7).

Part of this work was done during visits to Caltech’s Department of Mathematics (Pasadena), and to the Erwin Schrödinger Institute (Vienna). I am grateful to both institutions. I also wish to thank G. Ancona, J.I. Burgos Gil, M. Cavicchi, F. Déglise, F. Ivorra, F. Lemma, J. Tilouine and A. Vezzani for useful discussions and comments, as well as the referee for her or his observations and suggestions concerning an earlier version of this article.

Conventions: We use the triangulated, ${\mathbb{Q}}$-linear categories $\mathop{\mathop{DM_{\text{{\tencyr B}},c}}\nolimits(X)}\nolimits$ of constructible Beilinson motives over $X$ [CD, Def. 15.1.1], indexed by schemes $X$ over $\mathop{{\bf Spec}}\nolimits{\mathbb{Q}}$, which are separated and of finite type. As in [CD], the symbol $\mathds{1}_{X}$ is used to denote the unit for the tensor product in $\mathop{\mathop{DM_{\text{{\tencyr B}},c}}\nolimits(X)}\nolimits$. We shall employ the full formalism of six operations developed in [loc. cit.]. The reader may choose to consult [Hé, Sect. 2] or [Wi4, Sect. 1] for concise presentations of this formalism.

Beilinson motives can be endowed with a canonical weight structure, thanks to the main results from [Hé] (see [Bo, Prop. 6.5.3] for the case $X=\mathop{{\bf Spec}}\nolimits k$, for a field $k$ of characteristic zero). We refer to it as the motivic weight structure. Following [Wi4, Def. 1.5], the category $\mathop{CHM(X)_{\mathbb{Q}}}\nolimits$ of Chow motives over $X$ is defined as the heart $\mathop{DM_{\text{{\tencyr B}},c}}\nolimits(X)_{w=0}$ of the motivic weight structure on $\mathop{\mathop{DM_{\text{{\tencyr B}},c}}\nolimits(X)}\nolimits$.

A scheme will be said to be nilregular if the underlying reduced scheme is regular in the usual sense.

1 Statement of the main result

In order to state our main result (Theorem 1.6), let us introduce the situation we are going to consider. The ${\mathbb{Q}}$-scheme $M^{K}$ is a Siegel threefold, and the Chow motive ${}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ over $M^{K}$ is associated to a dominant weight $\mathop{\underline{\alpha}}\nolimits=(k_{1},k_{2},r)\in{\mathbb{Z}}^{3}$, $k_{1}\geq k_{2}\geq 0$ (see below for the precise normalizations). Denote by $j$ the open immersion of $M^{K}$ into its Satake–(Baily–Borel) compactification $(M^{K})^{*}$, and by $i:\partial(M^{K})^{*}\hookrightarrow(M^{K})^{*}$ the immersion of the complement of $M^{K}$ in $(M^{K})^{*}$ (with the reduced scheme structure, say). Recall the following.

Definition 1.1** (cmp. [Wi9, Def. 2.1 (a)]).**

Let $\mathop{CHM(M^{K})}\nolimits_{{\mathbb{Q}},\partial w\neq 0,1}$ denote the full sub-category of $\mathop{CHM(M^{K})}\nolimits_{\mathbb{Q}}$ of objects $V$ such that $i^{*}j_{*}V$ is without weights [math] and $1$.

Theorem 1.6 implies that in our setting, the motive ${}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}\in\mathop{CHM(M^{K})}\nolimits_{\mathbb{Q}}$ belongs to $\mathop{CHM(M^{K})}\nolimits_{{\mathbb{Q}},\partial w\neq 0,1}$ if and only if $\mathop{\underline{\alpha}}\nolimits$ is regular: $k_{1}>k_{2}>0$. More precisely, putting $k:=\min(k_{1}-k_{2},k_{2})$, the motive $i^{*}j_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ is without weights $-k+1,-k+2,\ldots,k$. The proof of Theorem 1.6 will be given in Section 2. It is an application of [Wi9, Thm. 4.4]; in order to verify the hypotheses of the latter, we heavily rely on results from [Lm].

Fix a four-dimensional ${\mathbb{Q}}$-vector space $V$, together with a ${\mathbb{Q}}$-valued non-degenerate symplectic bilinear form $J$.

Definition 1.2**.**

The group scheme $G$ over ${\mathbb{Q}}$ is defined as the group of symplectic similitudes

[TABLE]

Thus, $G$ is reductive, and for any ${\mathbb{Q}}$-algebra $R$, the group $G(R)$ equals

[TABLE]

In particular, the similitude norm $\lambda(g)$ defines a canonical morphism

[TABLE]

The group $G$ is split over ${\mathbb{Q}}$, and its center $Z(G)$ equals $\mathop{{\mathbb{G}}_{m,{\mathbb{Q}}}}\nolimits\subset\mathop{\rm GL}\nolimits(V)$ (inclusion of scalar automorphisms). Maximal ${\mathbb{Q}}$-split tori, together with an inclusion into a Borel sub-group of $G$, are in bijection with symplectic ${\mathbb{Q}}$-bases of $V$, in which $J$ acquires the $4\times 4$-matrix

[TABLE]

equally denoted by $J$. Here as in the sequel, we denote by $I_{2}$ the $2\times 2$-matrix representing the identity. Fix one such basis $(e_{1},e_{2},e_{3},e_{4})$, use it to identify $G$ with the sub-group $GSp_{4,{\mathbb{Q}}}$ of $GL_{4,{\mathbb{Q}}}$ of matrices $g$ satisfying the relation

[TABLE]

the maximal split torus with the sub-group $T$ of diagonal matrices

[TABLE]

and the Borel sub-group with the sub-group of matrices stabilizing the flag of totally isotropic sub-spaces $(e_{1})_{\mathbb{Q}}\subset(e_{1},e_{2})_{\mathbb{Q}}$ of $V$. We consider triplets $(k_{1},k_{2},r)\in{\mathbb{Z}}^{3}$ satisfying the congruence relation

[TABLE]

To such a triplet, let us associate the (representation-theoretic) weight

[TABLE]

Note that restriction of $\alpha(k_{1},k_{2},r)$ to $T\cap Sp(V,J)$ corresponds to the projection onto $(k_{1},k_{2})$. In particular, the weight $\alpha(k_{1},k_{2},r)$ is dominant if and only if $k_{1}\geq k_{2}\geq 0$; it is regular if and only if $k_{1}>k_{2}>0$. Note also that the composition of $\alpha(k_{1},k_{2},r)$ with the cocharacter

[TABLE]

equals

[TABLE]

The character $\lambda$ on $T$ equals $\alpha(0,0,-2)$, and $\det=\lambda^{2}$.

Definition 1.3**.**

The analytic space ${\cal H}$ is defined as the sub-space of $M_{2}({\mathbb{C}})$ of those complex $2\times 2$-matrices, which are symmetrical, and whose imaginary part is (positive or negative) definite:

[TABLE]

The group of real points $G({\mathbb{R}})$ acts on ${\cal H}$ by analytical automorphisms [P1, Ex. 2.7]. In fact, $(G,{\cal H})$ are pure Shimura data [P1, Def. 2.1]. Their reflex field [P1, Sect. 11.1] equals ${\mathbb{Q}}$. Given that $Z(G)=\mathop{{\mathbb{G}}_{m,{\mathbb{Q}}}}\nolimits$, the Shimura data $(G,{\cal H})$ satisfy condition $(+)$ from [Wi2, Sect. 5].

Let us now fix additional data:

1. (A)

an open compact sub-group $K$ of $G({\mathbb{A}}_{f})$ which is neat [P1, Sect. 0.6], 2. (B)

a triplet $(k_{1},k_{2},r)\in{\mathbb{Z}}^{3}$ satisfying the above congruence

[TABLE]

and in addition,

[TABLE]

In other words, the character $\mathop{\underline{\alpha}}\nolimits:=\alpha(k_{1},k_{2},r)$ is dominant.

These data are used as follows. The Shimura variety $M^{K}:=M^{K}(G,{\cal H})$ is smooth over ${\mathbb{Q}}$. This is the Siegel threefold of level $K$. According to [P1, Thm. 11.16], it admits an interpretation as modular space of Abelian surfaces with additional structures. In particular, there is a universal family ${\rm{B}}$ of Abelian surfaces over $M^{K}$.

The following result holds in the general context of (smooth) Shimura varieties of $PEL$-type.

Theorem 1.4** ([A, Thm. 8.6]).**

There is a ${\mathbb{Q}}$-linear tensor functor

[TABLE]

from the Tannakian category $\mathop{\rm Rep}\nolimits(G)$ of algebraic representations of $G$ in finite dimensional ${\mathbb{Q}}$-vector spaces to the ${\mathbb{Q}}$-linear category $CHM^{s}(M^{K})_{\mathbb{Q}}$ of smooth Chow motives over $M^{K}$ [Lv1, Def. 5.16]. It has the following properties.

1. (a)

The composition of $\widetilde{\mu}$ with the cohomological Hodge theoretic realization is isomorphic to the canonical construction functor $\mu_{\bf H}$ (e.g. **[Wi1, Thm. 2.2]**) to the category of admissible graded-polarizable variations of Hodge structure on $M^{K}_{\mathbb{C}}$. 2. (b)

The composition of $\widetilde{\mu}$ with the cohomological $\ell$-adic realization is isomorphic to the canonical construction functor $\mu_{\ell}$ (e.g. **[Wi1, Chap. 4]**) to the category of lisse $\ell$-adic sheaves on $M^{K}$. 3. (c)

The functor $\widetilde{\mu}$ commutes with Tate twists. 4. (d)

The functor $\widetilde{\mu}$ maps the representation $V$ to the dual of the Chow motive $\pi_{*}^{1}\mathds{1}_{{\rm{B}}}$ over $M^{K}$.

Here, we denote by $\pi_{*}^{m}\mathds{1}_{{\rm{B}}}$ the $m$-th Chow-Künneth component of the Chow motive $\pi_{*}\mathds{1}_{{\rm{B}}}$ over $M^{K}$ [DM, Thm. 3.1].

Proof of Theorem 1.4.  Parts (a), (c) and (d) are identical to [A, Thm. 8.6].

As for part (b), repeat the proof of [loc. cit.], observing that the $\ell$-adic analogue of [A, Prop. 8.5] holds (the base change to ${\mathbb{Q}}_{\ell}$ of the sub-group $G_{1}$ of $G$ coincides with the Lefschetz group).     q.e.d.

Given that the representation on $V$ is faithful, it follows that any object in the image of $\widetilde{\mu}$ is isomorphic to a direct sum of direct factors of Tate twists of the Chow motive $\pi_{n_{i},*}\mathds{1}_{{\rm{B}}^{n_{i}}}$ associated to ${\rm{B}}^{n_{i}}$, for suitable $n_{i}\in{\mathbb{N}}$, where $\pi_{n_{i}}:{\rm{B}}^{n_{i}}\to M^{K}$ denotes the $n_{i}$-fold fibre product of ${\rm{B}}$ over $M^{K}$.

Definition 1.5**.**

(a) Denote by $V_{\mathop{\underline{\alpha}}\nolimits}\in\mathop{\rm Rep}\nolimits(G)$ the irreducible representation of highest weight $\mathop{\underline{\alpha}}\nolimits$.

(b) Define ${}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}\in\mathop{CHM^{s}(M^{K})}\nolimits_{\mathbb{Q}}\subset\mathop{CHM(M^{K})}\nolimits_{\mathbb{Q}}$ as

[TABLE]

Given that $V_{\mathop{\underline{\alpha}}\nolimits}$ is of weight $r$, the cohomological realizations of ${}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ equal zero in (classical, i.e., non-perverse) degrees $\neq r$, and $\mu_{\bf H}(V_{\mathop{\underline{\alpha}}\nolimits})$ (in the Hodge theoretic setting) resp. $\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})$ (in the $\ell$-adic setting) in degree $r$.

Denote by $j:M^{K}\hookrightarrow(M^{K})^{*}$ the open immersion of $M^{K}$ into its Satake–(Baily–Borel) compactification, by $i:\partial(M^{K})^{*}\hookrightarrow(M^{K})^{*}$ its complement, and by $\Phi$ the natural stratification of $\partial(M^{K})^{*}$ (the latter will be made explicit in the beginning of Section 2). Here is our main result.

Theorem 1.6**.**

*(a) The motive $i^{*}j_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}\in\mathop{DM_{\text{{\tencyr B}},c}}\nolimits(\partial(M^{K})^{*})$ is a $\Phi$-constructible motive of Abelian type over $\partial(M^{K})^{*}$ (see Definition 2.1).

(b) The motive $i^{*}j_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ is without weights*

[TABLE]

where $k:=\min(k_{1}-k_{2},k_{2})$. Both weights $-k$ and $k+1$ do occur in $i^{*}j_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$. In particular, ${}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ belongs to the sub-category $\mathop{CHM(M^{K})}\nolimits_{{\mathbb{Q}},\partial w\neq 0,1}$ of $\mathop{CHM(M^{K})}\nolimits_{\mathbb{Q}}$ if and only if $\mathop{\underline{\alpha}}\nolimits$ is regular.

Theorem 1.6 should be compared to [Wi5, Thm. 3.5], [Wi6, Thm. 3.8], and [C, Thm. 3.6, Prop. 3.8, Prop. 3.9] (see also [Wi9, Rem. 5.8 (b)]), which treat the cases of Hilbert–Blumenthal varieties, of Picard surfaces, and of Picard varieties of arbitrary dimension, respectively.

Theorem 1.6 will be proved in Section 2. For the rest of the present section, assume that $k=\min(k_{1}-k_{2},k_{2})\geq 1$, i.e., $k_{1}>k_{2}>0$. Given that according to Theorem 1.6 (b), the motive ${}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ belongs to $\mathop{CHM(M^{K})}\nolimits_{{\mathbb{Q}},\partial w\neq 0,1}\,$, the intersection motive of $M^{K}$ relative to $(M^{K})^{*}$ with coefficients in ${}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ is at our disposal: by definition [Wi9, Def. 3.7], it equals

[TABLE]

where $a:(M^{K})^{*}\to\mathop{{\bf Spec}}\nolimits{\mathbb{Q}}$ is the structure morphism of $(M^{K})^{*}$. By abuse of language, let us abbreviate, and refer to $a_{*}\mathop{j_{!*}}\nolimits{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ as the intersection motive with coefficients in ${}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$. Let us list the main corollaries of Theorem 1.6.

Corollary 1.7**.**

*Denote by $a$ and $\tilde{a}$ the structure morphisms of $(M^{K})^{*}$ and $M^{K}$, respectively, and by $m$ the natural transformation $j_{!}\to j_{*}$. Assume $k_{1}>k_{2}>0$, i.e., $k\geq 1$.

(a) The motive $\tilde{a}_{!}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}\in\mathop{\mathop{DM_{\text{{\tencyr B}},c}}\nolimits({\mathbb{Q}})}\nolimits$ is without weights $-k,-k+1,\ldots,-1$, and the motive $\tilde{a}_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}\in\mathop{\mathop{DM_{\text{{\tencyr B}},c}}\nolimits({\mathbb{Q}})}\nolimits$ is without weights $1,2,\ldots,k$. More precisely, the exact triangles*

[TABLE]

and

[TABLE]

*are weight filtrations (of $\tilde{a}_{!}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$) avoiding weights $-k,-k+1,\ldots,-1$, and (of $\tilde{a}_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$) avoiding weights $1,2,\ldots,k$, respectively.

(b) The intersection motive $a_{*}\mathop{j_{!*}}\nolimits{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}\in\mathop{CHM({\mathbb{Q}})_{\mathbb{Q}}}\nolimits$ behaves functorially with respect to both $\tilde{a}_{!}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ and $\tilde{a}_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$. In particular, any endomorphism of $\tilde{a}_{!}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ or of $\tilde{a}_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ induces an endomorphism of $a_{*}\mathop{j_{!*}}\nolimits{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$.

(c) Let $\tilde{a}_{!}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}\to N\to\tilde{a}_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ be a factorization of the morphism $a_{*}m:\tilde{a}_{!}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}\to\tilde{a}_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ through a Chow motive $N\in\mathop{CHM({\mathbb{Q}})_{\mathbb{Q}}}\nolimits$. Then the intersection motive $a_{*}\mathop{j_{!*}}\nolimits{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ is canonically identified with a direct factor of $N$, with a canonical direct complement.*

Proof.  Given Theorem 1.6, parts (a), (b) and (c) follow from [Wi9, Thm. 3.4], [Wi9, Thm. 3.5] and [Wi3, Cor. 2.5], respectively.     q.e.d.

The equivariance statement from Corollary 1.7 (b) applies in particular to endomorphisms coming from the Hecke algebra ${\mathfrak{H}}(K,G({\mathbb{A}}_{f}))$ associated to the neat open compact sub-group $K$ of $G({\mathbb{A}}_{f})$. Recall that by what was said earlier, the relative Chow motive ${}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ is a direct factor of a Tate twist of $\pi_{N,*}\mathds{1}_{{\rm{B}}^{N}}$, where $\pi_{N}:{\rm{B}}^{N}\to M^{K}$ denotes the $N$-fold fibre product of the universal Abelian scheme ${\rm{B}}$ over $M^{K}$.

Corollary 1.8**.**

Assume $k\geq 1$. Then every element of the Hecke algebra ${\mathfrak{H}}(K,G({\mathbb{A}}_{f}))$ acts naturally on the intersection motive $a_{*}\mathop{j_{!*}}\nolimits{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$.

Proof.  Let $T\in{\mathfrak{H}}(K,G({\mathbb{A}}_{f}))$. According to Corollary 1.7 (b), it suffices to show that $T$ acts on $\tilde{a}_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$.

To do so, we refer to [Wi7, pp. 591–592].     q.e.d.

Corollary 1.9**.**

Assume $k\geq 1$, and let $\widetilde{{\rm{B}}^{N}}$ be any smooth compactification of ${\rm{B}}^{N}$. Then the intersection motive $a_{*}\mathop{j_{!*}}\nolimits{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ is a direct factor of a Tate twist of the Chow motive $b_{*}\mathds{1}_{\widetilde{{\rm{B}}^{N}}}$ ($b:=$ the structure morphism of the ${\mathbb{Q}}$-scheme $\widetilde{{\rm{B}}^{N}}$).

Proof.  The motive ${}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ is a direct factor of a Tate twist of $\pi_{N,*}\mathds{1}_{{\rm{B}}^{N}}$:

[TABLE]

for a suitable integer $\ell$. The morphism

[TABLE]

factors through the Chow motive $b_{*}\mathds{1}_{\widetilde{{\rm{B}}^{N}}}$, hence so does

[TABLE]

Now apply Corollary 1.7 (c).     q.e.d.

Remark 1.10**.**

When $r\geq 0$, then according to [A, Lemma 4.13], the Chow motive ${}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ is a direct factor of $\pi_{N,*}\mathds{1}_{{\rm{B}}^{N}}$ (no Tate twist needed).

In this context, let us recall [Wi9, Cor. 3.10]: the intersection motive $a_{*}\mathop{j_{!*}}\nolimits{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ is canonically dual to the $e_{\mathop{\underline{\alpha}}\nolimits}$-part of the interior motive of ${\rm{B}}^{N}$, where $e_{\mathop{\underline{\alpha}}\nolimits}$ is the idempotent endomorphism corresponding to the direct factor ${}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ of $\pi_{N,*}\mathds{1}_{{\rm{B}}^{N}}$.

Corollary 1.11**.**

Assume $k\geq 1$, i.e., that $\mathop{\underline{\alpha}}\nolimits$ is regular. Then for all $n\in{\mathbb{Z}}$, the natural maps

[TABLE]

(in the Hodge theoretic setting) and

[TABLE]

(in the $\ell$-adic setting) are injective. Dually,

[TABLE]

and

[TABLE]

are surjective. In other words, the natural maps from intersection cohomology to cohomology with coefficients in $\mu_{\bf H}(V_{\mathop{\underline{\alpha}}\nolimits})$, resp. $\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})$ identify intersection and interior cohomology.

Proof.  Write ${}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ as a direct factor of $\pi_{N,*}\mathds{1}_{{\rm{B}}^{N}}(\ell)[2\ell]$, for a suitable integer $\ell$. Given Theorem 1.6, we may quote [Wi9, Rem. 3.13 (a), (b)] (for $X=(M^{K})^{*}$, $U=M^{K}$, $C={\rm{B}}^{N}$ and $e=e_{\mathop{\underline{\alpha}}\nolimits}$).     q.e.d.

As pointed out in [Wi9, Rem. 3.13 (c)], sheaf theoretic considerations alone suffice to show (without any further reference to geometry) that Theorem 1.6 implies Corollary 1.11.

Corollary 1.11 is already known. Indeed, according to [MT, Prop. 1], the result generalizes to Siegel varieties of arbitrary dimension. (However, the proof of [loc. cit.] is analytic.)

Remark 1.12**.**

By [Wi3, Thm. 4.14], control of the reduction of some compactification of ${\rm{B}}^{N}$ implies control of certain properties of the $\ell$-adic realization of the intersection motive $a_{*}\mathop{j_{!*}}\nolimits{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$. According to [FC, Thm. VI.1.1], there exists a smooth compactification of ${\rm{B}}^{N}$ having good reduction at each prime number $p$ not dividing the level $n$ of $K$.

[Wi3, Thm. 4.14] then yields the following: (a) for all prime numbers $p$ not dividing $n$, the $p$-adic realization of $a_{*}\mathop{j_{!*}}\nolimits{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ is crystalline, (b) if furthermore $p$ and $\ell$ are different, then the $\ell$-adic realization of $a_{*}\mathop{j_{!*}}\nolimits{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ is unramified at $p$.

Corollary 1.13**.**

Assume $k\geq 1$. Let $p$ be a prime number not dividing the level of $K$. Let $\ell$ be different from $p$. Then the characteristic polynomials of the following coincide: (1) the action of Frobenius $\phi$ on the $\phi$-filtered module associated to the (crystalline) $p$-adic realization of the intersection motive $a_{*}\mathop{j_{!*}}\nolimits{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$, (2) the action of a geometrical Frobenius automorphism at $p$ on the (unramified) $\ell$-adic realization of $a_{*}\mathop{j_{!*}}\nolimits{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$.

Proof.  Fix a smooth compactification $\widetilde{{\rm{B}}^{N}}$ of ${\rm{B}}^{N}$ with good reduction at $p$ [FC, Thm. VI.1.1]. Thus, the ${\mathbb{Q}}_{p}$-scheme $\widetilde{{\rm{B}}^{N}}\times_{\mathbb{Q}}{\mathbb{Q}}_{p}$ is the generic fibre of a smooth and proper scheme $\widetilde{{\cal B}^{N}}$ over ${\mathbb{Z}}_{p}$. Let us denote by $\widetilde{{\cal B}^{N}_{{\mathbb{F}}_{p}}}$ its special fibre.

The $\phi$-filtered module associated to $p$-adic étale cohomology of $\widetilde{{\rm{B}}^{N}}\times_{\mathbb{Q}}\bar{{\mathbb{Q}}}$ is first isomorphic to Hyodo–Kato cohomology $H^{{}^{\mathchoice{\bullet}{\bullet}{\scriptscriptstyle\bullet}{\bullet}}}_{HK}(\widetilde{{\rm{B}}^{N}}\times_{\mathbb{Q}}\bar{{\mathbb{Q}}}_{p})$ [Be, Sect. 3.2], and this isomorphism can be chosen to be motivic in the sense that it commutes with the action of correspondences in $\widetilde{{\rm{B}}^{N}}\times_{\mathbb{Q}}\widetilde{{\rm{B}}^{N}}$ [DN, Sect. 4.15]. By definition, Hyodo–Kato cohomology is log-crystalline cohomology of a log-smooth model; in our case, given good reduction, such a model is given by $\widetilde{{\cal B}^{N}}$ (with divisor equal to zero). In other words, Hyodo–Kato cohomology equals crystalline cohomology of $\widetilde{{\cal B}^{N}}$. This identification commutes with the action of correspondences in $\widetilde{{\cal B}^{N}}\times_{{\mathbb{Z}}_{p}}\widetilde{{\cal B}^{N}}$. Finally, crystalline cohomology of $\widetilde{{\cal B}^{N}}$ equals crystalline cohomology of $\widetilde{{\cal B}^{N}_{{\mathbb{F}}_{p}}}$.

Altogether, the $\phi$-filtered module associated to $p$-adic étale cohomology of $\widetilde{{\rm{B}}^{N}}\times_{\mathbb{Q}}\bar{{\mathbb{Q}}}$ is identified with crystalline cohomology of $\widetilde{{\cal B}^{N}_{{\mathbb{F}}_{p}}}$ in a way compatible with the action of correspondences in $\widetilde{{\cal B}^{N}}\times_{{\mathbb{Z}}_{p}}\widetilde{{\cal B}^{N}}$. Concretely, this means that given a correspondence $e$ in $\widetilde{{\cal B}^{N}}\times_{{\mathbb{Z}}_{p}}\widetilde{{\cal B}^{N}}$, the action of its generic fibre $e_{{\mathbb{Q}}_{p}}$ on $p$-adic étale cohomology is identified with the action of its special fibre $e_{{\mathbb{F}}_{p}}$ on crystalline cohomology.

For $\ell\neq p$, smooth and proper base change allows us to identify $\ell$-adic cohomology of $\widetilde{{\rm{B}}^{N}}\times_{\mathbb{Q}}\bar{{\mathbb{Q}}}$ and $\ell$-adic cohomology of $\widetilde{{\cal B}^{N}_{{\mathbb{F}}_{p}}}\times_{{\mathbb{F}}_{p}}\bar{{\mathbb{F}}}_{p}$, again compatibly with correspondences.

According to Corollary 1.9, there is an idempotent endomorphism $e_{{\mathbb{Q}}}$ of the Chow motive associated to $\widetilde{{\rm{B}}^{N}}$, in other words, an idempotent correspondence in $\widetilde{{\rm{B}}^{N}}\times_{\mathbb{Q}}\widetilde{{\rm{B}}^{N}}$, whose images in the endomorphism rings of the realizations are projectors onto the realizations of $a_{*}\mathop{j_{!*}}\nolimits{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$. We claim that $e_{{\mathbb{Q}}_{p}}:=e_{\mathbb{Q}}\times_{\mathbb{Q}}{\mathbb{Q}}_{p}$ can be extended idempotently to $\widetilde{{\cal B}^{N}}\times_{{\mathbb{Z}}_{p}}\widetilde{{\cal B}^{N}}$. Indeed, according to [O’S, Prop. 5.1.1], the restriction morphism from the endomorphism ring of the Chow motive associated to $\widetilde{{\cal B}^{N}}$ to the one of the Chow motive associated to $\widetilde{{\rm{B}}^{N}}$ is epimorphic, with nilpotent kernel. We now follow a standard line of argument (cmp. [K, proof of Cor. 7.8]): let ${\mathfrak{e}}$ be any extension of $e_{{\mathbb{Q}}_{p}}$ to $\widetilde{{\cal B}^{N}}\times_{{\mathbb{Z}}_{p}}\widetilde{{\cal B}^{N}}$. The difference ${\mathfrak{e}}-{\mathfrak{e}}^{2}$ is nilpotent, say

[TABLE]

But then,

[TABLE]

equally extends $e_{{\mathbb{Q}}_{p}}$ to $\widetilde{{\cal B}^{N}}\times_{{\mathbb{Z}}_{p}}\widetilde{{\cal B}^{N}}$. and $e_{{\mathbb{Z}}_{p}}$ is idempotent. Altogether, there is a smooth and proper scheme $\widetilde{{\cal B}^{N}_{{\mathbb{F}}_{p}}}$ over ${\mathbb{F}}_{p}$, and an idempotent endomorphism $e_{{\mathbb{F}}_{p}}$ of the Chow motive associated to $\widetilde{{\cal B}^{N}_{{\mathbb{F}}_{p}}}$, whose images in the endomorphism rings of crystalline and $\ell$-adic cohomology, respectively, are projectors onto the realizations of $a_{*}\mathop{j_{!*}}\nolimits{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$. The claim thus follows from [KM, Thm. 2. 2)].     q.e.d.

2 Proof of the main result

We keep the notation of the preceding section. In order to prove Theorem 1.6, the idea is to apply the criterion from [Wi9, Cor. 4.6].

In order to check the hypotheses of [loc. cit.], we first need to fix a finite stratification $\Phi$ of $\partial(M^{K})^{*}$ by locally closed sub-schemes. The canonical choice would be the restriction $\Phi^{\prime}$ to $\partial(M^{K})^{*}$ of the natural (finite) stratification of $(M^{K})^{*}$ from [P1, Main Theorem 12.3 (c)], in other words, all the strata of $(M^{K})^{*}$ except the open one, i.e., except $M^{K}$. According to [Wi7, Lemma 8.2 (a)], $\Phi^{\prime}$ is good, meaning that the closure of every stratum is a union of strata. Furthermore [Wi7, Lemma 8.2 (b)], all strata, denoted $i_{g}(M^{\pi_{1}(K_{1})})$, are smooth over ${\mathbb{Q}}$ (recall that $K$ is assumed neat, and that $(G,{\cal H})$ satisfy condition $(+)$), hence regular. The same is therefore true for the following coarser stratification $\Phi=\{0,1\}$ of $\partial(M^{K})^{*}$: denote by $i_{0}:Z_{0}\hookrightarrow\partial(M^{K})^{*}$ the disjoint union of all closed strata of $\Phi^{\prime}$, and by $i_{1}:Z_{1}\hookrightarrow\partial(M^{K})^{*}$ the disjoint union of all strata of $\Phi^{\prime}$, which are open in $\partial(M^{K})^{*}$. Indeed, according to [P1, Sect. 6.3, Ex. 4.25 (with $g=2$)],

[TABLE]

more precisely, $Z_{0}$ is of dimension zero, and $Z_{1}$ of dimension one (hence so is the whole of $\partial(M^{K})^{*}$). Let us refer to $Z_{0}$ as the Siegel stratum, and to $Z_{1}$ as the Klingen stratum of $\partial(M^{K})^{*}$. When it will be necessary to insist on the structure of stratified scheme of $\partial(M^{K})^{*}$, we shall write $\partial(M^{K})^{*}(\Phi)$ instead of $\partial(M^{K})^{*}$.

Definition 2.1** ([Wi8, Def. 3.4 and 3.5]).**

(a) Let $S({\mathfrak{S}})=\coprod_{\sigma\in{\mathfrak{S}}}\mathop{S_{\sigma}}\nolimits$ be a good stratification of a scheme $S({\mathfrak{S}})$. A morphism $\pi:S({\mathfrak{S}})\to\partial(M^{K})^{*}(\Phi)$ is said to be a morphism of good stratifications if the pre-image of any of the strata $Z_{0}$, $Z_{1}$ of $\partial(M^{K})^{*}$ is a union of strata $\mathop{S_{\sigma}}\nolimits\,$.

(b) A morphism $\pi:S({\mathfrak{S}})\to\partial(M^{K})^{*}(\Phi)$ of good stratifications is said to be of Abelian type if it is proper, and if the following conditions are satisfied.

1. (1)

All strata $\mathop{S_{\sigma}}\nolimits$, $\sigma\in{\mathfrak{S}}$, are nilregular, and for any immersion $i_{\tau}:\mathop{S_{\tau}}\nolimits\hookrightarrow\mathop{\overline{\mathop{S_{\sigma}}\nolimits}}\nolimits$ of a stratum $\mathop{S_{\tau}}\nolimits$ into the closure $\mathop{\overline{\mathop{S_{\sigma}}\nolimits}}\nolimits$ of a stratum $\mathop{S_{\sigma}}\nolimits$, the functor $i_{\tau}^{!}$ maps $\mathds{1}_{\mathop{\overline{\mathop{S_{\sigma}}\nolimits}}\nolimits}$ to a Tate motive over $S_{\tau}$ [Lv2, Sect. 3.3]. 2. (2)

For all $\sigma\in{\mathfrak{S}}$ such that $\mathop{S_{\sigma}}\nolimits$ is a stratum of $\pi^{-1}(Z_{m})$, $m\in\{0,1\}$, the morphism $\mathop{\pi_{\sigma}}\nolimits:\mathop{S_{\sigma}}\nolimits\to Z_{m}$ can be factorized,

[TABLE]

such that the motive

[TABLE]

belongs to the category $\mathop{DMT(\mathop{B_{\sigma}}\nolimits)_{F}}\nolimits$ of Tate motives over $\mathop{B_{\sigma}}\nolimits\,$, the morphism $\mathop{\pi^{\prime}_{\sigma}}\nolimits$ is proper and smooth, and its pull-back to any geometric point of $Z_{m}$ lying over a generic point is isomorphic to a finite disjoint union of Abelian varieties.

(c) An object $V\in\mathop{DM_{\text{{\tencyr B}},c}}\nolimits(\partial(M^{K})^{*})$ is a $\Phi$-constructible motive of Abelian type over $\partial(M^{K})^{*}$ if the following holds: the motive $V$ belongs to the strict, full, dense, ${\mathbb{Q}}$-linear triangulated sub-category $\mathop{\mathop{DM_{\text{{\tencyr B}},c,\Phi}^{Ab}}\nolimits(\partial(M^{K})^{*})}\nolimits$ generated by the images under $\pi_{*}$ of ${\mathfrak{S}}$-constructible Tate motives over $S({\mathfrak{S}})$ [Wi8, Def. 3.3], where

[TABLE]

runs through the morphisms of Abelian type with target equal to $\partial(M^{K})^{*}(\Phi)$.

Theorem 2.2**.**

Let $\mathop{\underline{\alpha}}\nolimits=\alpha(k_{1},k_{2},r)$, with $(k_{1},k_{2},r)\in{\mathbb{Z}}^{3}$ such that

[TABLE]

and consider ${}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}=\widetilde{\mu}(V_{\mathop{\underline{\alpha}}\nolimits})\in\mathop{CHM(M^{K})}\nolimits_{\mathbb{Q}}\,$. The motive $i^{*}j_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ belongs to the full sub-category $\mathop{\mathop{DM_{\text{{\tencyr B}},c,\Phi}^{Ab}}\nolimits(\partial(M^{K})^{*})}\nolimits$ of $\mathop{DM_{\text{{\tencyr B}},c}}\nolimits(\partial(M^{K})^{*})\,$. In other words, it is a $\Phi$-constructible motive of Abelian type over $\partial(M^{K})^{*}$.

Proof.  As recalled earlier, the relative Chow motive ${}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ belongs to the strict, full, dense, ${\mathbb{Q}}$-linear triangulated sub-category

[TABLE]

of $\mathop{DM_{\text{{\tencyr B}},c}}\nolimits(M^{K})$ generated by the images under $\pi_{N,*}$ of the category of Tate motives over ${\rm{B}}^{N}$. Here as before, $\pi_{N}:{\rm{B}}^{N}\to M^{K}$ denotes the $N$-fold fibre product of the universal Abelian scheme ${\rm{B}}$ over $M^{K}$.

The latter equals the projection from a mixed Shimura variety: indeed [P1, Ex. 2.7], the representation $V$ of $G$ is of Hodge type $\{(-1,0),(0,-1)\}$. The same is then true for the $r$-th power $V^{N}$ of $V$. By [P1, Prop. 2.17], this allows for the construction of the unipotent extension $(P^{N},{\mathfrak{X}}^{N})$ of $(G,{\cal H})$ by $V^{N}$. The pair $(P^{N},{\mathfrak{X}}^{N})$ constitute mixed Shimura data [P1, Def. 2.1]. By construction, they come endowed with a morphism $\pi_{N}:(P^{N},{\mathfrak{X}}^{N})\to(G,{\cal H})$ of Shimura data, identifying $(G,{\cal H})$ with the pure Shimura data underlying $(P^{N},{\mathfrak{X}}^{N})$. In particular, $(P^{N},{\mathfrak{X}}^{N})$ also satisfy condition $(+)$. Now [P1, Thm. 11.18] there is an open compact neat subgroup $K_{N}$ of $P^{N}({\mathbb{A}}_{f})$, whose image under $\pi_{N}$ equals $K$, such that ${\rm{B}}^{N}$ is identified with the mixed Shimura variety $M^{K_{N}}:=M^{K_{N}}(P^{N},{\mathfrak{X}}^{N})$, and such the morphism $M^{K_{N}}\to M^{K}$ induced by the morphism $\pi_{N}$ of Shimura data is identified with the structure morphism of ${\rm{B}}^{N}$.

Choose a smooth toroidal compactification $M^{K_{N}}({\mathfrak{S}}):=M^{K_{N}}(P^{N},{\mathfrak{X}}^{N},{\mathfrak{S}})$ of $M^{K_{N}}$, associated to a $K_{N}$-admissible complete cone decomposition ${\mathfrak{S}}$ [P1, Sect. 6.4]. Then [P1, proof of Thm. 9.21], modulo a suitable refinement of ${\mathfrak{S}}$, the natural stratification of $M^{K_{N}}({\mathfrak{S}})$, also denoted ${\mathfrak{S}}$, satisfies the conclusions of [Wi7, Lemma 8.1], i.e., it is good, and the closures of all strata are regular. Note that the unique open stratum equals $M^{K_{N}}$. According to [P1, Sect. 6.24, Main Theorem 12.4 (b)], the morphism $\pi_{N}:{\rm{B}}^{N}=M^{K_{N}}\to M^{K}$ extends to a proper, surjective morphism $M^{K_{N}}({\mathfrak{S}})\to(M^{K})^{*}$, still denoted $\pi_{N}$. From the description given in [P1, Sect. 7.3], one sees that $\pi_{N}$ is a morphism of stratifications.

According to [Wi7, Cor. 4.10 (b), Rem. 4.7], the category

[TABLE]

is obtained by gluing $\pi_{N,*}DMT({\rm{B}}^{N})_{\mathbb{Q}}^{\natural}$ and $\pi_{N,*}DMT_{\mathfrak{S}}(\pi_{N}^{-1}(\partial(M^{K})^{*}))_{\mathbb{Q}}^{\natural}\,$. In particular,

[TABLE]

But $\pi_{N}$ is of Abelian type [Wi7, Lemma 8.4]; therefore,

[TABLE]

q.e.d.

Next, we collect information on the restriction of $i^{*}j_{*}R_{\ell,M^{K}}({}^{\mathop{\underline{\alpha}}\nolimits}{\cal V})$ to the strata $Z_{0}$ and $Z_{1}$. The following is essentially due to Lemma [Lm, Sect. 4].

Theorem 2.3**.**

*Let $\ell$ be a prime number.

(a) For all integers $n\leq r+2$, the perverse cohomology sheaf*

[TABLE]

on $Z_{0}$ is of weights $\leq n-(k_{1}-k_{2})$. The perverse cohomology sheaf

[TABLE]

*is non-zero, and pure of weight $(r+2)-(k_{1}-k_{2})$.

(b) For all integers $n\leq r+2$, the perverse cohomology sheaf*

[TABLE]

on $Z_{1}$ is of weights $\leq n-k_{2}$. The perverse cohomology sheaf

[TABLE]

is non-zero, and pure of weight $(r+2)-k_{2}$.

The proof of Theorem 2.3 will be given after Remark 2.6. In order to prepare it, recall [P1, Ex. 4.25] that $Z_{0}$ and $Z_{1}$ correspond bijectively to the $G({\mathbb{Q}})$-conjugacy classes of proper rational boundary components [P1, Sect. 4.11] of $(G,{\cal H})$. Indeed, the group $G({\mathbb{Q}})$ acts transitively on the set of totally isotropic sub-spaces of $V$ of a given, strictly positive dimension.

We already fixed a basis $(e_{1},e_{2},e_{3},e_{4})$ of $V$, in which our symplectic bilinear form $J$ acquires the $4\times 4$-matrix

[TABLE]

which we equally denoted by $J$. The sub-spaces $V_{0}^{\prime}$ and $V_{1}^{\prime}$ generated by $\{e_{1},e_{2}\}$ and $\{e_{1}\}$, respectively, are both totally isotropic.

Following [P1, Ex. 4.25], we put $Q_{m}:=Stab_{G}(V_{m}^{\prime})$, $m=0,1$. Let $P_{m}$ denote the normal sub-group of $Q_{m}$ underlying the rational boundary component $(P_{m},{\mathfrak{X}}_{m})$ giving rise to $Z_{m}$ [P1, Sect. 4.7], and $W_{m}$ its unipotent radical (which equals the unipotent radical of $Q_{m}$). Then, still according to [P1, Ex. 4.25], $Q_{0}$ equals

[TABLE]

[TABLE]

[TABLE]

while $Q_{1}$ equals

[TABLE]

[TABLE]

[TABLE]

Observe that $Q_{0}\cap Q_{1}$ equals the Borel sub-group of $G$ stabilizing the flag $V_{1}^{\prime}\subset V_{0}^{\prime}$, and that both $Q_{0}$ and $Q_{1}$ contain the fixed maximal split torus

[TABLE]

In particular, $T$ is canonically identified with a maximal ${\mathbb{Q}}$-split torus of the reductive group $Q_{m}/W_{m}$, for $m=0,1$. Given a (representation-theoretic) weight $\alpha:T\to\mathop{{\mathbb{G}}_{m,{\mathbb{Q}}}}\nolimits$, let us denote by $\alpha_{m}$ the same application, but with $T$ seen as a sub-group of $Q_{m}/W_{m}$, $m=0,1$.

Note that

[TABLE]

Recall that we denote by $\Phi^{\prime}$ the natural (finite) stratification of $(M^{K})^{*}$ from [P1, Main Theorem 12.3 (c)], which is finer than $\Phi$. In order to determine the classical cohomology objects $R^{n}i_{m}^{*}i^{*}j_{*}\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})$, for $m=0,1$, and $n\in{\mathbb{Z}}$, one applies the following standard strategy. (1) By Pink’s Theorem [P2, Thm. (5.3.1)], the restriction of $R^{n}i_{m}^{*}i^{*}j_{*}\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})$ to any individual stratum $Z^{\prime}$ of $\Phi^{\prime}$ contributing to $Z_{m}$ equals

[TABLE]

Here, $H_{C}/K_{W}$ is an arithmetic sub-group (depending on $Z^{\prime}$) of $C_{m}/W_{m}$ [P2, Sect. (5.2)], where $C_{m}$ is the identity component of the Zariski closure of the centralizer in $Q_{m}({\mathbb{Q}})$ of the rational boundary component $(P_{m},{\mathfrak{X}}_{m})$ [P2, Sect. (3.7)], and $\mu_{\ell,Z^{\prime}}$ is the canonical construction functor to the category of lisse $\ell$-adic sheaves on $Z^{\prime}$. (2) Apply Kostant’s Theorem [V, Thm. 3.2.3], in order to identify $H^{q}(Lie(W_{m}),V_{\mathop{\underline{\alpha}}\nolimits})$ as a representation of the reductive group $Q_{m}/W_{m}$; this allows in particular to obtain its weights, and gives potential information concerning cohomology of $H_{C}/K_{W}$ with coeffients in $H^{q}(Lie(W_{m}),V_{\mathop{\underline{\alpha}}\nolimits})$.

The Hodge theoretic analogue of the above strategy yields the cohomology objects of $i_{m}^{*}i^{*}j_{*}\mu_{\bf H}(V_{\mathop{\underline{\alpha}}\nolimits})_{|Z^{\prime}}\,$; this was made explicit in [Lm, Sect. 4]. Since steps (2) of the $\ell$-adic and the Hodge theoretic strategies are identical, we may use the computations from [loc. cit.] in our setting.

Proposition 2.4** ([Lm, Sect. 4.3]).**

Let $\mathop{\underline{\alpha}}\nolimits=\alpha(k_{1},k_{2},r)$, with $(k_{1},k_{2},r)\in{\mathbb{Z}}^{3}$ such that

[TABLE]

(a) For $m=0,1$, we have

[TABLE]

*whenever $q<0$ or $q>3$. If $0\leq q\leq 3$, then the $Q_{m}/W_{m}$-representation $H^{q}(Lie(W_{1}),V_{\mathop{\underline{\alpha}}\nolimits})$ is (non-zero and) irreducible.

(b) The highest (representation-theoretic) weight of $H^{q}(Lie(W_{0}),V_{\mathop{\underline{\alpha}}\nolimits})$, $0\leq q\leq 3$, is*

[TABLE]

[TABLE]

[TABLE]

[TABLE]

(c) The highest (representation-theoretic) weight of $H^{q}(Lie(W_{1}),V_{\mathop{\underline{\alpha}}\nolimits})$, $0\leq q\leq 3$, is

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proof.  Note that given our normalization, we have

[TABLE]

in the notation of [Lm, top of p. 87].

Part (a) follows from Kostant’s Theorem, and from the following fact (see [Lm, proof of Lemma 4.8] and [Lm, proof of Lemma 4.10]), valid for both $m=0$ and $m=1$: the set of Weyl representatives for $Q_{m}$ contains no element of length $<0$ or $>3$, and exactly one element of respective lengths [math], $1$, $2$ and $3$.

As for part (c), we refer to [Lm, proof of Lemma 4.10].

[Lm, proof of Lemma 4.8] contains the complete setting for the application of Kostant’s Theorem for $m=0$, but makes it explicit only for $H^{2}(Lie(W_{0}),V_{\mathop{\underline{\alpha}}\nolimits})$ and $H^{3}(Lie(W_{0}),V_{\mathop{\underline{\alpha}}\nolimits})$. The reader will have no difficulty to fill in the missing information needed for part (b).     q.e.d.

Note that both $Q_{0}/W_{0}$ and $Q_{1}/W_{1}$ are isomorphic to the direct product $\mathop{{\mathbb{G}}_{m,{\mathbb{Q}}}}\nolimits\times_{\mathbb{Q}}GL_{2,{\mathbb{Q}}}$. More precisely,

[TABLE]

the identification given by sending the class of a matrix

[TABLE]

to the pair $(q,A)$, and

[TABLE]

the identification given by sending the class of a matrix

[TABLE]

to the pair

[TABLE]

The restriction of the inverse identification to maximal split tori sends

[TABLE]

to

[TABLE]

for $m=0$, and

[TABLE]

to

[TABLE]

for $m=1$.

In the following, the reader will be particularly careful not to confuse two notions of weight associated to representations of reductive groups: the highest weights in the sense of representation theory (e.g., those occurring in Kostant’s Theorem), when the representation is irreducible, and the weights as determined by the action of the weight cocharacter [P1, Sect. 1.3], when the group underlies Shimura data.

Corollary 2.5**.**

(a) The $Q_{0}/W_{0}$-representations $H^{q}(Lie(W_{0}),V_{\mathop{\underline{\alpha}}\nolimits})$, $0\leq q\leq 2$, are (irreducible and) regular, except when $q=0$ and $k_{1}=k_{2}$, in which case $H^{0}(Lie(W_{0}),V_{\mathop{\underline{\alpha}}\nolimits})$ factors through the quotient $\mathop{{\mathbb{G}}_{m,{\mathbb{Q}}}}\nolimits\times_{\mathbb{Q}}\mathop{{\mathbb{G}}_{m,{\mathbb{Q}}}}\nolimits$ of the group

[TABLE]

via* the determinant on the factor $GL_{2,{\mathbb{Q}}}$. The restriction to $SL_{2,{\mathbb{Q}}}\subset GL_{2,{\mathbb{Q}}}$ of $H^{1}(Lie(W_{0}),V_{\mathop{\underline{\alpha}}\nolimits})$ is of highest (representation-theoretic) weight $k_{1}+k_{2}+2$. The restriction to $P_{0}/W_{0}$ of $H^{0}(Lie(W_{0}),V_{\mathop{\underline{\alpha}}\nolimits})$ is of weight $(r+1)-(k_{1}+k_{2})-1$, and the restriction of $H^{1}(Lie(W_{0}),V_{\mathop{\underline{\alpha}}\nolimits})$ is of weight $(r+2)-(k_{1}-k_{2})$.

(b) The restriction to $P_{1}/W_{1}$ of $H^{0}(Lie(W_{1}),V_{\mathop{\underline{\alpha}}\nolimits})$ is of weight $(r+1)-k_{1}-1$, and the restriction of $H^{1}(Lie(W_{1}),V_{\mathop{\underline{\alpha}}\nolimits})$ is of weight $(r+2)-k_{2}-1$.*

Proof.  (a): Given the above identifications, the weight $\alpha_{0}(n_{1},n_{2},r)$ on $T$ maps

[TABLE]

to

[TABLE]

In particular, the restriction of $\alpha_{0}(n_{1},n_{2},r)$ to $T\cap SL_{2,{\mathbb{Q}}}$ corresponds to the integer $n_{1}-n_{2}$. The first and the second claim thus follow from Proposition 2.4 (b).

The weight cocharacter $\mathop{{\mathbb{G}}_{m,{\mathbb{Q}}}}\nolimits\to P_{0}/W_{0}=\mathop{{\mathbb{G}}_{m,{\mathbb{Q}}}}\nolimits$ maps $z$ to $z^{2}$ [P1, Ex. 4.25, Ex. 2.8]. Its composition with the inclusion into $T$, and with $\alpha_{0}(n_{1},n_{2},r)$ yields

[TABLE]

The third claim thus follows from Proposition 2.4 (b), and from the normalization of weights of representations [P1, Sect. 1.3].

(b): The weight cocharacter $\mathop{{\mathbb{G}}_{m,{\mathbb{Q}}}}\nolimits\to P_{1}/W_{1}=GL_{2,{\mathbb{Q}}}$ maps $z$ to

[TABLE]

[P1, Ex. 4.25, Ex. 2.8]. Given the above identifications, its composition with the inclusion into $T$ maps $z$ to $\mathop{\rm diag}\nolimits(z^{2},z,1,z)$. Further composition with $\alpha_{1}(n_{1},n_{2},r)$ then yields

[TABLE]

The claim thus follows from Proposition 2.4 (c).     q.e.d.

In order to complete the ingredients needed for the computation of the $R^{n}i_{m}^{*}i^{*}j_{*}\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})$ according to the strategy (1), (2) sketched ealier in this section, observe that the group $H_{C}/K_{W}$ associated to an individual stratum $Z^{\prime}$ of $\Phi^{\prime}$ contributing to $Z_{m}$ is a neat arithmetic sub-group of $GL_{2}({\mathbb{Q}})$ for $m=0$ [Lm, proof of Lemma 4.8], hence of $SL_{2}({\mathbb{Q}})$. In particular, it is of cohomological dimension one. For $m=1$, the group $H_{C}/K_{W}$, being a neat arithmetic sub-group of ${\mathbb{G}}_{m}({\mathbb{Q}})$, is trivial [Lm, proof of Lemma 4.10].

Remark 2.6**.**

When $m=0$, let $V_{2}$ denote the standard representation of $SL_{2,{\mathbb{Q}}}$, and $u\in{\mathbb{N}}$. Then $\mathop{\rm Sym}\nolimits^{u}V_{2}\in\mathop{\rm Rep}\nolimits(SL_{2,{\mathbb{Q}}})$; in fact, $\mathop{\rm Sym}\nolimits^{u}V_{2}$ is the irreductible representation of highest (representation-theoretic) weight $u$. Denote by $g$ the genus of the quotient of the upper half space by $H_{C}/K_{W}$, and by $c\geq 1$ the number of its cusps. (Thus, $c\geq 3$ if $g=0$ since $H_{C}/K_{W}$ is neat.) Then $H^{1}(H_{C}/K_{W},\mathop{\rm Sym}\nolimits^{u}V_{2})$ is of dimension $(u+1)(2g-2+c)$ if $u\geq 1$, and of dimension $2g-1+c$ if $u=0$. In particular,

[TABLE]

Proof of Theorem 2.3.  (a): According to Corollary 2.5 (a) and Proposition 2.4 (a),

1. (o)

$0\neq H^{0}(Lie(W_{0}),V_{\mathop{\underline{\alpha}}\nolimits})$ is of weight $(r+1)-(k_{1}+k_{2})-1$, 2. (i)

$0\neq H^{1}(Lie(W_{0}),V_{\mathop{\underline{\alpha}}\nolimits})$ is of weight $(r+2)-(k_{1}-k_{2})$,

and $H^{q}(Lie(W_{0}),V_{\mathop{\underline{\alpha}}\nolimits})=0$ whenever $q<0$. The group $H_{C}/K_{W}$ associated to a stratum $Z^{\prime}$ of $Z_{0}$ is a neat arithmetic sub-group of $SL_{2}({\mathbb{Q}})$. It is therefore of cohomological dimension one, and admits no non-zero invariants on regular irreducible representations of $Q_{0}/W_{0}=\mathop{{\mathbb{G}}_{m,{\mathbb{Q}}}}\nolimits\times_{\mathbb{Q}}GL_{2,{\mathbb{Q}}}$.

According to Proposition 2.4 (a) and Corollary 2.5 (a), $H^{q}(Lie(W_{0}),V_{\mathop{\underline{\alpha}}\nolimits})$, $0\leq q\leq 2$, are irreducible as representations of $Q_{0}/W_{0}$, and regular unless $q=0$ and $k_{1}=k_{2}$, in which case $SL_{2,{\mathbb{Q}}}$, hence $H_{C}/K_{W}$ acts trivially. Pink’s Theorem and [P2, Prop. (5.5.4)] then tell us that

1. (o)

$R^{0}i_{0}^{*}i^{*}j_{*}\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})$ is non-zero if and only if $k_{1}=k_{2}$, in which case it is of weight $r-(k_{1}+k_{2})$, 2. (i)

$0\neq R^{1}i_{0}^{*}i^{*}j_{*}\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})$ is of weight $(r+1)-(k_{1}+k_{2})-1$, 3. (ii)

$0\neq R^{2}i_{0}^{*}i^{*}j_{*}\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})$ is of weight $(r+2)-(k_{1}-k_{2})$,

and that $R^{n}i_{0}^{*}i^{*}j_{*}\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})=0$ whenever $n<0$ (for the non-vanishing statements in (i), (ii), see Remark 2.6).

The scheme $Z_{0}$ is of dimension zero; therefore,

[TABLE]

From (o), (i), (ii) and the vanishing of $R^{n}i_{0}^{*}i^{*}j_{*}\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})=0$ for $n<0$, we conclude that

1. (r)

$H^{r}i_{0}^{*}i^{*}j_{*}R_{\ell,M^{K}}({}^{\mathop{\underline{\alpha}}\nolimits}{\cal V})$ is zero if $k_{1}>k_{2}$, and non-zero of weight $r-(k_{1}+k_{2})$ if $k_{1}=k_{2}$, 2. (r+1)

$0\neq H^{r+1}i_{0}^{*}i^{*}j_{*}R_{\ell,M^{K}}({}^{\mathop{\underline{\alpha}}\nolimits}{\cal V})$ is of weight $(r+1)-(k_{1}+k_{2})-1$, 3. (r+2)

$0\neq H^{r+2}i_{0}^{*}i^{*}j_{*}R_{\ell,M^{K}}({}^{\mathop{\underline{\alpha}}\nolimits}{\cal V})$ is of weight $(r+2)-(k_{1}-k_{2})$,

and that $H^{n}i_{0}^{*}i^{*}j_{*}R_{\ell,M^{K}}({}^{\mathop{\underline{\alpha}}\nolimits}{\cal V})=0$ whenever $n\leq r-1$.

(b): According to Corollary 2.5 (b) and Proposition 2.4 (a),

1. (o)

$0\neq H^{0}(Lie(W_{1}),V_{\mathop{\underline{\alpha}}\nolimits})$ is of weight $(r+1)-k_{1}-1$, 2. (i)

$0\neq H^{1}(Lie(W_{1}),V_{\mathop{\underline{\alpha}}\nolimits})$ is of weight $(r+2)-k_{2}-1$,

and $H^{q}(Lie(W_{1}),V_{\mathop{\underline{\alpha}}\nolimits})=0$ whenever $q<0$. The group $H_{C}/K_{W}$ associated to a stratum $Z^{\prime}$ of $Z_{1}$ is trivial. Pink’s Theorem and [P2, Lemma (5.6.6)] then tell us that

1. (o)

$0\neq R^{0}i_{1}^{*}i^{*}j_{*}\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})$ is of weight $(r+1)-k_{1}-1$, 2. (i)

$0\neq R^{1}i_{1}^{*}i^{*}j_{*}\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})$ is of weight $(r+2)-k_{2}-1$,

and that $R^{n}i_{1}^{*}i^{*}j_{*}\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})=0$ whenever $n<0$. Furthermore, Pink’s Theorem tells us that all classical cohomology objects $R^{n}i_{1}^{*}i^{*}j_{*}\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})$, $n\in{\mathbb{Z}}$ are lisse. The formula

[TABLE]

is valid: the first equation comes from

[TABLE]

As for the second, note that any lisse $\ell$-adic sheaf ${\cal F}$ on a one-dimensional regular scheme is a perverse sheaf ${\cal F}^{\prime}$ up to a shift by $-1$:

[TABLE]

From (o), (i) and the vanishing of $R^{n}i_{1}^{*}i^{*}j_{*}\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})=0$ for $n<0$, we conclude that

1. (r+1)

$0\neq H^{r+1}i_{1}^{*}i^{*}j_{*}R_{\ell,M^{K}}({}^{\mathop{\underline{\alpha}}\nolimits}{\cal V})$ is of weight $(r+1)-k_{1}$, 2. (r+2)

$0\neq H^{r+2}i_{1}^{*}i^{*}j_{*}R_{\ell,M^{K}}({}^{\mathop{\underline{\alpha}}\nolimits}{\cal V})$ is of weight $(r+2)-k_{2}$,

and that $H^{n}i_{1}^{*}i^{*}j_{*}R_{\ell,M^{K}}({}^{\mathop{\underline{\alpha}}\nolimits}{\cal V})=0$ whenever $n\leq r$.     q.e.d.

For the final step of the proof of Theorem 1.6, the following commutative diagram of immersions will be useful.

[TABLE]

Immersions situated on the same line are complementary to each other (example: $j^{\prime\prime}$ and $i^{\prime\prime}$), the four immersions marked by “$\circ$” are open (example: $i_{1}$), and the other four are closed (example: $i^{\prime}$).

Remark 2.7**.**

Denote by $\tau^{t\leq{\mathchoice{\bullet}{\bullet}{\scriptscriptstyle\bullet}{\bullet}}}_{Z_{m}}$ and $\tau^{t\geq{\mathchoice{\bullet}{\bullet}{\scriptscriptstyle\bullet}{\bullet}}}_{Z_{m}}$ the truncation functors with respect to the perverse $t$-structure on $Z_{m}$, $m=0,1$.

(a) The immersions $j^{\prime}$ and $i^{\prime}$ being complementary,

[TABLE]

for any perverse sheaf ${\cal F}^{\prime}$ on $M^{K}$ [BBD, Prop. 1.4.23].

(b) The intermediate extension is transitive, i.e.,

[TABLE]

[BBD, Cor. 1.4.24]. Application of the functor $(i^{\prime\prime})^{*}j_{*}^{\prime\prime}$ to the exact triangle

[TABLE]

of functors on perverse sheaves on $M^{K}$ (see (a)) yields the exact triangle

[TABLE]

The immersions $j^{\prime\prime}$ and $i^{\prime\prime}$ being complementary, we have as in (a)

[TABLE]

for any perverse sheaf ${\cal F}^{\prime\prime}$ on $(M^{K})^{*}-Z_{0}$. It follows that for any perverse sheaf ${\cal F}^{\prime}$ on $M^{K}$, there are exact sequences of perverse cohomology objects

[TABLE]

[TABLE]

for $n\leq-1$, while $H^{n}(i_{0}^{*}i^{*}\mathop{j_{!*}}\nolimits{\cal F}^{\prime})=0$ for all $n\geq 0$.

(c) Recall that $R_{\ell,M^{K}}({}^{\mathop{\underline{\alpha}}\nolimits}{\cal V})=\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})[-r]$; the variety $M^{K}$ being of dimension three, the complex $R_{\ell,M^{K}}({}^{\mathop{\underline{\alpha}}\nolimits}{\cal V})$ is therefore concentrated in perverse degree $r+3$. According to our conventions, $i_{1}^{*}i^{*}\mathop{j_{!*}}\nolimits R_{\ell,M^{K}}({}^{\mathop{\underline{\alpha}}\nolimits}{\cal V})=(i^{\prime})^{*}\mathop{j_{!*}^{\prime}}\nolimits R_{\ell,M^{K}}({}^{\mathop{\underline{\alpha}}\nolimits}{\cal V})$ thus equals

[TABLE]

According to (a), we thus have

[TABLE]

Similarly, following (b),

[TABLE]

for all $n\geq r+3$, and there are exact sequences

[TABLE]

[TABLE]

for $n\leq r+2$.

(d) We claim that

[TABLE]

for all $n\leq r+1$. Equivalently,

[TABLE]

for all $n\leq 1$. Indeed, by Pink’s Theorem, the classical cohomology objects of $i_{1}^{*}i^{*}j_{*}\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})$ are all lisse. Applying $\tau^{t\geq 3}_{Z_{1}}$, we thus get a complex concentrated in classical degrees $\geq 2$ (recall that $Z_{1}$ is of dimension one). The same is thus true after application of $i_{0}^{*}i_{1,*}$ (recall that inverse images are $t$-exact for the classical $t$-structure). In other words, the complex

[TABLE]

has trivial cohomology (classical or perverse; recall that $Z_{0}$ is of dimension zero) in degrees $\leq 1$.

(e) From (c) and (d), we deduce that

[TABLE]

for $n\leq r+1$, and that $H^{r+2}i_{0}^{*}i^{*}\mathop{j_{!*}}\nolimits R_{\ell,M^{K}}({}^{\mathop{\underline{\alpha}}\nolimits}{\cal V})$ equals the kernel of

[TABLE]

Corollary 2.8**.**

*Let $\ell$ be a prime number.

(a) For all $n\in{\mathbb{Z}}$,*

[TABLE]

*is of weights $\leq n-(k_{1}-k_{2})$.

(b) For all $n\in{\mathbb{Z}}$,*

[TABLE]

is of weights $\leq n-k_{2}$. The perverse cohomology sheaf

[TABLE]

is non-zero, and pure of weight $(r+2)-k_{2}$.

Proof.  Part (a) follows from Remark 2.7 (c), (e), and from Theorem 2.3 (a).

Part (b) follows from Remark 2.7 (c), and from Theorem 2.3 (b).     q.e.d.

Corollary 2.8 suffices to prove the part of Theorem 1.6 (b) asserting that regularity of $\mathop{\underline{\alpha}}\nolimits$ is sufficient for weights [math] and $1$ to be avoided by $i^{*}j_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$. In order to prove that it is necessary, we need the following statement.

Proposition 2.9**.**

Let $\ell$ be a prime number. Then provided that $k_{1}\geq 1$, the perverse cohomology sheaf

[TABLE]

is non-zero, and pure of weight $(r+2)-(k_{1}-k_{2})$.

Proof.  According to Remark 2.7 (e),

[TABLE]

equals the kernel of

[TABLE]

— in particular, it is pure of weight $(r+2)-(k_{1}-k_{2})$ (Theorem 2.3 (a)) —, i.e., it equals the kernel of

[TABLE]

Thanks to Pink’s Theorem, the regularity of $H^{2}(Lie(W_{0}),V_{\mathop{\underline{\alpha}}\nolimits})$ as a representation of $Q_{0}/W_{0}$ (Corollary 2.5 (a)), and the fact that the group $H_{C}/K_{W}$ is of cohomological dimension one, locally on $Z_{0}$, the (perverse or classical) sheaf

[TABLE]

equals

[TABLE]

for a stratum $Z^{\prime}$ of $\Phi^{\prime}$ contributing to $Z_{0}$. Furthermore (Corollary 2.5 (a)), the restriction of $H^{1}(Lie(W_{0}),V_{\mathop{\underline{\alpha}}\nolimits})$ to the group $H_{C}/K_{W}$ is isomorphic to the $(k_{1}+k_{2}+2)$-nd symmetric power of the standard representation of $SL_{2,{\mathbb{Q}}}$. By Remark 2.6, $H^{2}i_{0}^{*}i^{*}j_{*}\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})_{|Z^{\prime}}$ is therefore of constant rank $(k_{1}+k_{2}+3)(2g-2+c)$, where $g$ denotes the genus of $H_{C}/K_{W}$, and $c$ the number of cusps.

We claim that the restriction to the same $Z^{\prime}$ of

[TABLE]

is of constant rank $c$. Indeed, according to Remark 2.7 (d), the classical cohomology objects of $i_{1}^{*}i^{*}j_{*}\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})$ are all lisse. Therefore, perverse truncation above degree three equals classical truncation above degree two (recall that $Z_{1}$ is of dimension one). The complex

[TABLE]

is concentrated in degrees $\geq 2$, and we get

[TABLE]

Restriction to $Z^{\prime}$ yields

[TABLE]

where the direct sum is indexed by all strata $Z^{\prime\prime}$ contributing to $Z_{1}$, and containing $Z^{\prime}$ in their closure. For every such $Z^{\prime\prime}$,

[TABLE]

according to Pink’s Theorem (since the group $H_{C}/K_{W}$ (for $m=1$!) is trivial).

Denote by $j_{1}:Z_{1}\hookrightarrow Z_{1}^{*}$ the Baily–Borel compactification, and by $i_{01}:\partial Z_{1}^{*}\hookrightarrow Z_{1}^{*}$ its complement. The immersion $i_{1}:Z_{1}\hookrightarrow(M^{K})^{*}$ admits a natural extension $\bar{i}_{1}:Z_{1}^{*}\to(M^{K})^{*}$ [P1, Main Thm. 12.3 (c), Sect. 7.6], which is finite. The diagram

[TABLE]

is cartesian up to nilpotent elements. Proper base change therefore yields the formula

[TABLE]

The functors $\bar{i}_{1,*}$ and $i_{0,1}^{*}$ being exact on sheaves, we have

[TABLE]

According to Proposition 2.4 (a), $H^{2}(Lie(W_{1}),V_{\mathop{\underline{\alpha}}\nolimits})$ is irreducible as a representation of $Q_{1}/W_{1}$, hence of $GL_{2,{\mathbb{Q}}}$. Yet another application of Pink’s Theorem shows that

[TABLE]

is of constant rank one on the intersection of $\partial Z_{1}^{*}$ with the closure of $Z^{\prime\prime}$ in $(Z_{1})^{*}$.

Our claim on the rank of

[TABLE]

is therefore proven as soon as we establish that the number of points in the geometrical fibres of the morphism $\bar{i}_{1}:\partial Z_{1}^{*}\to Z_{0}$ above $Z^{\prime}\subset Z_{0}$ equals $c$. This verification can be done on the level of ${\mathbb{C}}$-valued points, where the adelic description of the situation is at our disposal. More precisely, write $(G_{m},{\cal H}_{m}):=(P_{m},{\mathfrak{X}}_{m})/W_{m}$ [P1, Prop. 2.9], $m=0,1$, for the Shimura data contributing to $\partial(M^{K})^{*}$, and $Q_{01}$ for the Borel sub-group $Q_{0}\cap Q_{1}$ of $G$. According to [P1, Sect. 6.3], the diagram of ${\mathbb{C}}$-valued points corresponding to the diagram

[TABLE]

equals

[TABLE]

where all maps are induced by canonical inclusions of groups and spaces. Indeed, the full group $Q_{m}({\mathbb{Q}})$ (and not only a sub-group of finite index) stabilizes ${\cal H}_{m}$, $m=0,1$, and two rational boundary components of $(G_{1},{\cal H}_{1})$ are conjugate under $G_{1}({\mathbb{Q}})$ if and only if they are conjugate under $G({\mathbb{Q}})$ (explicit computation, or [P1, (iii) of Remark on p. 91]). The sub-scheme $Z^{\prime}\subset Z_{0}$ equals the image of a Shimura variety associated to $(G_{0},{\cal H}_{0})$ under a morphism $i_{g}$ associated to an element $g\in G({\mathbb{A}}_{f})$ [P1, Main Theorem 12.3 (c)]; given the adelic description of $i_{g}$ from [P1, Sect. 6.3], we see that under the above identification, any $z\in Z^{\prime}({\mathbb{C}})$ equals the class $[h_{0},p_{0}g]$ in

[TABLE]

of a pair of the form $(h_{0},p_{0}g)$, with $h_{0}\in{\cal H}_{0}$ and $p_{0}\in P_{0}({\mathbb{A}}_{f})$. Put

[TABLE]

this group equals the centralizer in $Q_{0}({\mathbb{Q}})$ of $h_{0}$, and indeed, of the whole of ${\cal H}_{0}$. Putting

[TABLE]

we leave it to the reader to verify that the map

[TABLE]

is well-defined, and bijective. By strong approximation,

[TABLE]

But

[TABLE]

meaning that modulo $W_{0}$, elements in $P_{0}$ and in $Q_{0}$ commute with each other. Thus,

[TABLE]

The image of $W_{0}({\mathbb{Q}})\cdot H_{C}^{\prime}$ under the projection \pi_{0}:Q_{0}\mathrel{\vbox{\offinterlineskip\hbox spread13.99995pt{\hfil\scriptstyle\ \hfil}\hbox to24.16661pt{\rightarrowfill\hskip-7.96677pt\rightarrow}}}Q_{0}/W_{0} coincides with the image of

[TABLE]

(both images equals $\pi_{0}(Q_{0}^{+}({\mathbb{Q}}))\cap\pi(gKg^{-1})$). But by definition [P2, (3.7.4)], $W_{0}({\mathbb{A}}_{f})\cdot Q_{0}^{+}({\mathbb{Q}})\cap gKg^{-1}$ equals $H_{C}$. We thus showed that

[TABLE]

Now the quotient morphism Q_{0}\mathrel{\vbox{\offinterlineskip\hbox spread13.99995pt{\hfil\scriptstyle\ \hfil}\hbox to24.16661pt{\rightarrowfill\hskip-7.96677pt\rightarrow}}}Q_{0}/P_{0},q_{0}\mapsto\overline{q_{0}} induces an isomorphism

[TABLE]

But $\overline{Q_{0}({\mathbb{Q}})}=GL_{2}({\mathbb{Q}})$, and under this identification, $\overline{Q_{01}({\mathbb{Q}})}$ equals the sub-group of upper triangular matrices, while $\overline{H_{C}}=H_{C}/K_{W}$. In other words,

[TABLE]

is identified with the set up cusps of $H_{C}/K_{W}$.

The formula

[TABLE]

(recall that $c$ is greater or equal to $1$, and that $c\geq 3$ if $g=0$) implies that the rank of the source of $ad$ is strictly greater than the rank of its target; the kernel of $ad$ is therefore non-trivial.     q.e.d.

Remark 2.10**.**

(a) As the reader may verify,

[TABLE]

is pure of weight $(r+2)-(k_{1}-k_{2})$, i.e., of the same weight as

[TABLE]

Weight considerations alone do therefore not imply non-triviality of the kernel of the map $ad$ from the proof of Proposition 2.9.

(b) A more conceptual proof of Proposition 2.9 would consist in showing that locally on $Z_{0}$, the map $ad$ equals the direct sum over all cusps of $H_{C}/K_{W}$ of the residue maps. Identify $H^{1}(H_{C}/K_{W},H^{1}(Lie(W_{0}),V_{\mathop{\underline{\alpha}}\nolimits}))\otimes_{{\mathbb{Q}}}{\mathbb{C}}$ with the direct sum of the space of modular forms and (the conjugate of) the space of cusp forms for $H_{C}/K_{W}$ of weight $k_{1}+k_{2}+4\geq 5$. The kernel of the residues contains the space of cusp forms. Its dimension is computed in [Sh, Thm. 2.24 and Thm. 2.25]; thanks to [Sh, Prop. 1.40] (always remember that $H_{C}/K_{W}$ is neat), this dimension can be seen to be strictly positive.

(c) On the level of geometry of Baily–Borel compactifications, a “strange duality” seems to be involved in the proof of Proposition 2.9: we need to know how many modular curves in the boundary of $(M^{K})^{*}$ contain a given cusp $Z^{\prime}$ in their closure. The response yields the number of cusps of a “modular curve”, which does not explicitely occur in $(M^{K})^{*}$, namely the quotient of the upper half space by $H_{C}/K_{W}$. It would be interesting to see how this phenomenon generalizes to higher dimensional Siegel varieties.

(d) Our computation of the fibres of the morphism $\bar{i}_{1}:Z_{1}^{*}\to(M^{K})^{*}$ over points of $Z_{0}$ is a quantitative version of a classical non-injectivity result of Satake [Sat, Exemple on p. 13-06].

Remark 2.11**.**

The Hodge theoretic analogues of Theorem 2.3, Corollary 2.8 and Proposition 2.9 hold. The proofs are identical up to the use of Pink’s Theorem, which is replaced by [BW, Thm. 2.9].

Proof of Theorem 1.6.  According to Theorem 2.2, $i^{*}j_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ is a $\Phi$-constructible motive of Abelian type over $\partial(M^{K})^{*}$; this proves part (a) of our claim.

By [P1, Summ. 1.18 (d)], there is a perfect pairing

[TABLE]

in $\mathop{\rm Rep}\nolimits(G)$.

Fix a prime $\ell$. Applying $\mu_{\ell}$, we get a perfect pairing

[TABLE]

of $\ell$-adic lisse sheaves on $M^{K}$. In terms of local duality, the pairing induces an isomorphism

[TABLE]

($M^{K}$ is smooth of dimension three). Given $R_{\ell,M^{K}}({}^{\mathop{\underline{\alpha}}\nolimits}{\cal V})=\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})[-r]$, we find that

[TABLE]

where $s=r+3$.

Corollary 2.8 tells us that for all $n\in{\mathbb{Z}}$, and $m=0,1$,

[TABLE]

is of weights $\leq n-k$. According to [Wi9, Cor. 4.6 (b)], the motive $i^{*}j_{*}{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ therefore avoids weights $-k+1,-k+2,\ldots,k$.

In order to conclude the proof of part (b), it remains to show, again thanks to [Wi9, Cor. 4.6 (b)], that for some $n\in{\mathbb{Z}}$, and $m=0$ or $m=1$, weight $n-k$ does occur in

[TABLE]

We take $n=r+2$, and distinguish two cases. If $k=k_{2}$, i.e., $k_{2}\leq k_{1}-k_{2}$, take $m=1$; the claim then follows from Corollary 2.8 (b). Else, $k_{2}>k_{1}-k_{2}$ and $k=k_{1}-k_{2}$. Since $k_{1}\geq k_{2}$, we necessarily have $k_{1}\geq 1$. Take $m=0$ and apply Proposition 2.9.     q.e.d.

Remark 2.12**.**

(a) An element of $H^{n}(\partial(M^{K})^{*}({\mathbb{C}}),i^{*}j_{*}\mu_{\bf H}(V_{\mathop{\underline{\alpha}}\nolimits}))$ is called a ghost class if it lies in the image of

[TABLE]

and in the kernel of both restriction maps

[TABLE]

$m=0,1$. One of the main results of [M] implies that if $\mathop{\underline{\alpha}}\nolimits$ is regular, then there are no non-zero ghost classes [M, Thm. 3.1]. This result does not formally imply, nor is it implied by our Theorem 1.6. Nonetheless, it might be worthwhile to note that the weight arguments that occur in the proofs are quite similar. The most relevant information from Theorem 1.6, as far as [M, Thm. 3.1] is concerned, comes from the weight filtration

[TABLE]

avoiding weights $1,2,\ldots,k$ (Corollary 1.7 (a)), hence avoiding weight $1$ if $\mathop{\underline{\alpha}}\nolimits$ is regular, which we assume in the sequel. This implies that any element of $H^{n}(M^{K}({\mathbb{C}}),\mu_{\bf H}(V_{\mathop{\underline{\alpha}}\nolimits}))$ not mapping to zero in $H^{n}(\partial(M^{K})^{*}({\mathbb{C}}),i^{*}j_{*}\mu_{\bf H}(V_{\mathop{\underline{\alpha}}\nolimits}))$, remains non-zero in

[TABLE]

In other words, a ghost class vanishing in $H^{n}(\partial(M^{K})^{*}({\mathbb{C}}),\tau_{\partial(M^{K})^{*}}^{t\geq 3}i^{*}j_{*}\mu_{\bf H}(V_{\mathop{\underline{\alpha}}\nolimits}))$ is zero. The Hodge structure $H^{n}(\partial(M^{K})^{*}({\mathbb{C}}),\tau_{\partial(M^{K})^{*}}^{t\geq 3}i^{*}j_{*}\mu_{\bf H}(V_{\mathop{\underline{\alpha}}\nolimits}))$ has weights $\geq(r+n)+2$; the same type of considerations as those leading to Corollary 2.8 then imply that the direct sum of the restriction maps

[TABLE]

$m=0,1$, is injective.

(b) The above illustrates an observation made by Moya Giusti: for a class in the cohomology of the boundary whose weight is neither the middle weight nor the middle weight plus one we can determine exactly whether it is or not in the image of the morphism

[TABLE]

In fact, it appears amusing to note that the “middle weight” is relevant in another context than the one studied in the present paper. According to [M, p. 2317, second paragraph], the representation $V_{\mathop{\underline{\alpha}}\nolimits}$ satisfies the middle weight property if the space of ghost classes in $H^{n}(\partial(M^{K})^{*}({\mathbb{C}}),i^{*}j_{*}\mu_{\bf H}(V_{\mathop{\underline{\alpha}}\nolimits}))$ is pure of weight $r+n$. [M, Thm. 3.1] implies in particular that for all $\mathop{\underline{\alpha}}\nolimits$ (regular or not), the representation $V_{\mathop{\underline{\alpha}}\nolimits}$ does satisfy the middle weight property, while our Theorem 1.6 implies that weights $\{r+n,r+n+1\}$ do not occur at all in $H^{n}(\partial(M^{K})^{*}({\mathbb{C}}),i^{*}j_{*}\mu_{\bf H}(V_{\mathop{\underline{\alpha}}\nolimits}))$, as soon as $\mathop{\underline{\alpha}}\nolimits$ is regular.

Remark 2.13**.**

Saper’s vanishing theorem [Sap, Thm. 5] says that if $\mathop{\underline{\alpha}}\nolimits$ is regular, then the groups $H^{n}(M^{K}({\mathbb{C}}),\mu_{\bf H}(V_{\mathop{\underline{\alpha}}\nolimits}))$, hence (by comparison) $H^{n}(M^{K}\times_{\mathbb{Q}}\bar{{\mathbb{Q}}},\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits}))$ vanish for $n<3=\dim M^{K}$. By duality, one obtains that $H^{n}_{c}(M^{K}({\mathbb{C}}),\mu_{\bf H}(V_{\mathop{\underline{\alpha}}\nolimits}))=0$ and $H^{n}_{c}(M^{K}\times_{\mathbb{Q}}\bar{{\mathbb{Q}}},\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits}))=0$ for $n>3$. It follows that interior cohomology with coefficients in $\mu_{\bf H}(V_{\mathop{\underline{\alpha}}\nolimits})$, denoted

[TABLE]

and interior cohomology with coefficients in $\mu_{\ell}(V_{\mathop{\underline{\alpha}}\nolimits})$, denoted

[TABLE]

both vanish for $n\neq 3$, provided that $\mathop{\underline{\alpha}}\nolimits$ is regular.

3 The motive for an automorphic form

This final section contains the analogues for Siegel threefolds of the main results from [Sc]. Since we shall not restrict ourselves to the case of Hecke eigenforms, our notation becomes a little more technical than in [loc. cit.].

We continue to consider the situation of Sections 1 and 2. In particular, we fix a dominant $\mathop{\underline{\alpha}}\nolimits=\alpha(k_{1},k_{2},r)$, which we assume to be regular, i.e., $k_{1}>k_{2}>0$. Consider the intersection motive $a_{*}\mathop{j_{!*}}\nolimits{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}\in\mathop{CHM({\mathbb{Q}})_{\mathbb{Q}}}\nolimits$, where as before $a:(M^{K})^{*}\to\mathop{{\bf Spec}}\nolimits{\mathbb{Q}}$ denotes the structure morphism of $(M^{K})^{*}$. According to [Wi9, Rem. 3.13 (a)] and Remark 2.13, its Hodge theoretic realization equals

[TABLE]

and its $\ell$-adic realization equals

[TABLE]

By Corollary 1.8, every element of the Hecke algebra ${\mathfrak{H}}(K,G({\mathbb{A}}_{f}))$ acts on $a_{*}\mathop{j_{!*}}\nolimits{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$.

Theorem 3.1** ([Ha, Thm. 3.1.1]).**

Let $L$ be any field of characteristic zero. Then the ${\mathfrak{H}}(K,G({\mathbb{A}}_{f}))\otimes_{\mathbb{Q}}L$-module $H^{3}_{!}(M^{K}({\mathbb{C}}),\mu_{\bf H}(V_{\mathop{\underline{\alpha}}\nolimits}))\otimes_{\mathbb{Q}}L$ is semi-simple.

Note that [Ha, Sect. 8.1.6, p. 232] gives a proof of Theorem 3.1, while the statement in [Ha, Thm. 3.1.1] is “non-adelic”. Denote by $R({\mathfrak{H}}):=R({\mathfrak{H}}(K,G({\mathbb{A}}_{f})))$ the image of the Hecke algebra in the endomorphism algebra of $H^{3}_{!}(M^{K}({\mathbb{C}}),\mu_{\bf H}(V_{\mathop{\underline{\alpha}}\nolimits}))$.

Corollary 3.2**.**

Let $L$ be any field of characteristic zero. Then the $L$-algebra $R({\mathfrak{H}})\otimes_{\mathbb{Q}}L$ is semi-simple.

In particular, the isomorphism classes of simple right $R({\mathfrak{H}})\otimes_{\mathbb{Q}}L$-modules correspond bijectively to isomorphism classes of minimal right ideals.

Fix $L$, and let $Y_{\pi_{f}}$ be such a minimal right ideal of $R({\mathfrak{H}})\otimes_{\mathbb{Q}}L$. There is a (primitive) idempotent $e_{\pi_{f}}\in R({\mathfrak{H}})\otimes_{\mathbb{Q}}L$ generating $Y_{\pi_{f}}$.

Definition 3.3**.**

(a) The Hodge structure $W(\pi_{f})$ associated to $Y_{\pi_{f}}$ is defined as

[TABLE]

(b) Let $\ell$ be a prime number. The Galois module $W(\pi_{f})_{\ell}$ associated to $Y_{\pi_{f}}$ is defined as

[TABLE]

Definition 3.3 (b) should be compared to [We, Thm. I].

Proposition 3.4**.**

There is canonical isomorphism of Hodge structures

[TABLE]

and a canonical isomorphism of Galois modules

[TABLE]

Proof.  We shall perform the proof for Hodge structures; the one for Galois modules is formally identical. Obviously,

[TABLE]

is canonically identified with

[TABLE]

by mapping an morphism $g$ to the image of $1=1_{R({\mathfrak{H}})}$ under $g$. Inside

[TABLE]

the object $W(\pi_{f})$ contains precisely those morphisms $g$ vanishing on $1-e_{\pi_{f}}$, in other words, satisfying the relation $g(1)=g(e_{\pi_{f}})=g(1)\cdot e_{\pi_{f}}$.     q.e.d.

Since we do not know whether the Chow motive $a_{*}\mathop{j_{!*}}\nolimits{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$ is finite dimensional, we cannot apply [K, Cor. 7.8], and therefore do not know whether $e_{\pi_{f}}$ can be lifted idempotently to the Hecke algebra ${\mathfrak{H}}(K,G({\mathbb{A}}_{f}))$. This is why we need to descend to the level of Grothendieck motives. Denote by $a_{*}\mathop{j_{!*}}\nolimits{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}^{\prime}$ the Grothendieck motive underlying $a_{*}\mathop{j_{!*}}\nolimits{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}$.

Definition 3.5**.**

Assume $\mathop{\underline{\alpha}}\nolimits=\alpha(k_{1},k_{2},r)$ to be regular. Let $L$ be a field of characteristic zero, and $Y_{\pi_{f}}$ a minimal right ideal of $R({\mathfrak{H}})\otimes_{\mathbb{Q}}L$. The motive associated to $Y_{\pi_{f}}$ is defined as

[TABLE]

Definition 3.5 should be compared to [Sc, Sect. 4.2.0]. Given our construction, the following is obvious.

Theorem 3.6**.**

Assume $\mathop{\underline{\alpha}}\nolimits=\alpha(k_{1},k_{2},r)$ to be regular, i.e., $k_{1}>k_{2}>0$. Let $L$ be a field of characteristic zero, and $Y_{\pi_{f}}$ a minimal right ideal of $R({\mathfrak{H}})\otimes_{F}L$. The realizations of the motive ${\cal W}(\pi_{f})$ associated to $Y_{\pi_{f}}$ are concentrated in the single cohomological degree $r+3$, and they take the values $W(\pi_{f})$ (in the Hodge theoretic setting) resp. $W(\pi_{f})_{\ell}$ (in the $\ell$-adic setting).

A special case occurs when $Y_{\pi_{f}}$ is of dimension one over $L$, i.e., corresponds to a non-trivial character of $R({\mathfrak{H}})$ with values in $L$. The automorphic form is then an eigenform for the Hecke algebra. This is the analogue of the situation considered in [Sc] for elliptic cusp forms.

The motive ${\cal W}(\pi_{f})$ being a direct factor of $a_{*}\mathop{j_{!*}}\nolimits{}^{\mathop{\underline{\alpha}}\nolimits}{\cal V}^{\prime}$, our results on the latter from Section 1 have obvious consequences for the realizations of ${\cal W}(\pi_{f})$.

Corollary 3.7**.**

*Assume $\mathop{\underline{\alpha}}\nolimits=\alpha(k_{1},k_{2},r)$ to be regular. Let $L$ be a field of characteristic zero, and $Y_{\pi_{f}}$ a minimal right ideal of $R({\mathfrak{H}})\otimes_{\mathbb{Q}}L$. Let $p$ be a prime number not dividing the level of $K$. Let $\ell$ be different from $p$.

(a) The $p$-adic realization $W(\pi_{f})_{p}$ of ${\cal W}(\pi_{f})$ is crystalline.

(b) The $\ell$-adic realization $W(\pi_{f})_{\ell}$ of ${\cal W}(\pi_{f})$ is unramified at $p$.

(c) The characteristic polynomials of the following coincide: (1) the action of Frobenius $\phi$ on the $\phi$-filtered module associated to $W(\pi_{f})_{p}$, (2) the action of a geometrical Frobenius automorphism at $p$ on $W(\pi_{f})_{\ell}$.*

Proof.  Parts (a) and (b) follow from Remark 1.12.

As for (c), in order to apply [KM, Thm. 2. 2)], use that both realizations are cut out by the same cycle from the cohomology of a smooth and proper scheme over the field ${\mathbb{F}}_{p}$ (cmp. the proof of Corollary 1.13).     q.e.d.

Corollary 3.7 should be compared to [Sc, Thm. 1.2.4].

Remark 3.8**.**

Part (c) of Corollary 3.7 is already contained in [U, Thm. 1].