Relative deformation theory, relative Selmer groups, and lifting irreducible Galois representations

TL;DR

This paper investigates conditions under which irreducible mod p Galois representations over totally real fields can be lifted to geometric p-adic representations, expanding understanding of their deformation theory and lifting properties.

Contribution

It establishes new criteria for geometric lifts of Galois representations and proves non-geometric lifting results without oddness assumptions.

Findings

Any locally liftable irreducible mod p Galois representation with certain local conditions admits a geometric p-adic lift.

Proves non-geometric lifting results for Galois representations without the oddness restriction.

Provides a framework connecting deformation theory, Selmer groups, and lifting of Galois representations.

Abstract

We study irreducible odd mod $p$ Galois representations $\bar{\rho} \colon \mathrm{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_p)$, for $F$ a totally real number field and $G$ a general reductive group. For $p \gg_{G, F} 0$, we show that any $\bar{\rho}$ that lifts locally, and at places above $p$ to de Rham and Hodge-Tate regular representations, has a geometric $p$-adic lift. We also prove non-geometric lifting results without any oddness assumption.

Equations

$$G({\overline{\mathbb{Z}}}_p)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\bar{\rho }:{\Gamma }_{\mathbb{Q}}\to {\mathrm{GL}}_2({\overline{\mathbb{F}}}_p)$$

$$h^0({\Gamma }_{F_v},\bar{\rho }({\mathfrak{g}}^{\mathrm{der}}))=\dim ({\mathrm{Flag}}_{G^0}),$$

$${\bar{\rho }}^c\cong {\bar{\rho }}^{\vee }\otimes \mu {|}_{{\Gamma }_F},$$

$$\bar{\rho }(cg{c}^{-1})=A{}^t\bar{\rho }(g{)}^{-1}{A}^{-1}\mu (g)$$

$$\bar{r}:{\Gamma }_{\mathbb{Q}}\to ({\mathrm{GL}}_n\times {\mathrm{GL}}_1)({\overline{\mathbb{F}}}_p)⋊\{1,j\},$$

$$G({\mathcal{O}}^{\prime })\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

We would like to thank Gebhard Boeckle, Prakash Belkale, Arvind Nair, Ravi Ramakrishna, Jack Thorne, and Akshay Venkatesh for useful conversations and correspondence. We are especially grateful to Ravi Ramakrishna for providing detailed comments on an earlier draft. We also thank the referees for their helpful reports, which have improved the exposition. N.F. was supported by the DAE, Government of India, PIC 12-R&D-TFR-5.01-0500. C.K. would like to thank TIFR, Mumbai for its hospitality, in periods when some of the work was carried out. S.P. was supported by NSF grants DMS-1700759, DMS-1752313, and DMS-2120325.

In memory of Jean-Pierre Wintenberger 1954–2019

1. 1 Introduction

2. 1.1 The lifting problem for odd representations

3. 1.2 Main theorem

1. Introduction

Let $\overline{\mathbb{Z}}_{p}$ be the integral closure of $\mathbb{Z}_{p}$ in $\overline{\mathbb{Q}}_{p}$, and let $G$ be a smooth group scheme over $\overline{\mathbb{Z}}_{p}$ such that $G^{0}$ is a split connected reductive group.

1.1. The lifting problem for odd representations

The starting point of this paper is the following basic question:

Question 1.1**.**

Let $F$ be a number field with algebraic closure $\overline{F}$ and absolute Galois group $\Gamma_{F}=\mathrm{Gal}(\overline{F}/F)$, and let $\bar{\rho}\colon\mathrm{Gal}(\overline{F}/F)\to G(\overline{\mathbb{F}}_{p})$ be a continuous homomorphism. Does there exist a lift $\rho$

[TABLE]

that is geometric in the sense of Fontaine–Mazur?

This question has attracted a great deal of attention, at least since Serre proposed his modularity conjecture ([serre:conjectures]). We begin by recalling a few instances of this general problem, beginning with Serre’s conjecture. Serre proposed that every irreducible representation

[TABLE]

that was moreover odd in the sense that $\det\bar{\rho}(c)=-1$ for any complex conjugation $c\in\Gamma_{\mathbb{Q}}$ should be isomorphic to the mod $p$ reduction of a $p$-adic Galois representation attached to a classical modular eigenform. In particular, such a $\bar{\rho}$ should admit a geometric $p$-adic lift. Around the time Serre first made his conjecture, as recounted in a letter of Serre to Tate on 12th July, 1974 ([serre-tate:correspondance]), Deligne raised the objection that the conjecture implied the existence of geometric lifts of $\bar{\rho}$ which were moreover minimally ramified (for example unramified outside $p$ if $\bar{\rho}$ is unramified outside $p$). The papers [khare-wintenberger:serre0], [khare:serrelevel1], [khare-wintenberger:serre1], [khare-wintenberger:serre2] prove Serre’s modularity conjecture, and as a key step lift $\bar{\rho}$ to a geometric representation with prescribed local properties. The proof of this key step uses the modularity lifting results of Wiles and Taylor ([wiles:fermat], [taylor-wiles:fermat]). In contrast, prior to the resolution of Serre’s conjecture, Ramakrishna ([ramakrishna:lifting], [ramakrishna02]) developed a beautiful, purely Galois-theoretic, method that in most cases settled Question 1.1 in the setting of Serre’s conjecture ($F=\mathbb{Q}$, $G=\mathrm{GL}_{2}$, $\bar{\rho}$ odd and irreducible). Ramakrishna’s lifts cannot be ensured to be minimally ramified.

We might then turn to asking Question 1.1 for $\bar{\rho}\colon\Gamma_{\mathbb{Q}}\to\mathrm{GL}_{2}(\overline{\mathbb{F}}_{p})$ that are even, in the sense that $\det(\bar{\rho}(c))=1$. For instance, suppose that the image of $\bar{\rho}$ is $\mathrm{SL}_{2}(\mathbb{F}_{p})$. Any geometric lift would (for $p\neq 2$) itself be even, and so conjecturally would be the $p$-adic representation $\rho$ attached to an algebraic Maass form. Such a $\rho$ should, up to twist, have finite image (because up to twist the associated motive should have Hodge realization of type $(0,0)$); but for $p>5$, Dickson’s classification of finite subgroups of $\mathrm{PGL}_{2}(\mathbb{C})$ rules out the possibility of such a lift. Thus one expects that $\bar{\rho}$ has no geometric lift. We have no general means of translating this conjectural heuristic into a proof, but Calegari ([calegari:even2, Theorem 5.1]) has given an ingenious argument that proves unconditionally that certain such even $\bar{\rho}$ have no geometric lift.

In other settings, Question 1.1 is even more mysterious. For instance, if $G=\mathrm{GL}_{2}$ and $F/\mathbb{Q}$ is quadratic imaginary, we do not even have a reliable heuristic for predicting whether $\bar{\rho}\colon\Gamma_{F}\to\mathrm{GL}_{2}(\overline{\mathbb{F}}_{p})$ should have a geometric lift! It is a remarkable and widely-tested phenomenon that torsion cohomology (Hecke eigen-) classes for the locally symmetric spaces associated to congruence subgroups of $\mathrm{GL}_{2}/F$ need not lift to characteristic zero; one can ask whether after raising the level (passing to a finite covering space of the arithmetic 3-manifold) their Hecke eigensystems lift, and that the corresponding Galois-theoretic statement holds as well. However, we have little evidence to support this.

This paper addresses Question 1.1 for general $G$, but for $\bar{\rho}$ that are odd in a sense generalizing Serre’s formulation for $\mathrm{GL}_{2}$. The following definition is essentially due to Gross ([gross:odd]), who suggested parallels between this class of Galois representations and the “odd” representations of Serre’s original conjecture:

Definition 1.2**.**

We say $\bar{\rho}\colon\Gamma_{F}\to G(\overline{\mathbb{F}}_{p})$ is odd if for all $v\mid\infty$,

[TABLE]

where $\mathfrak{g}^{\mathrm{der}}$ is the Lie algebra of the derived group $G^{\mathrm{der}}$ of $G^{0}$, and $\mathrm{Flag}_{G^{0}}$ is the flag variety of $G^{0}$.

Note that for any involution of $\mathfrak{g}^{\mathrm{der}}$, the dimension of the space of invariants must be at least $\dim(\mathrm{Flag}_{G^{0}})$. An adjoint group contains an order 2 element whose invariants have dimension $\dim(\mathrm{Flag}_{G^{0}})$ if and only if $-1$ belongs to the Weyl group of $G$. When $-1$ does not belong to the Weyl group, we can (after choosing a pinning) find such an order two element in $G\rtimes\operatorname{Out}(G)$; for more details, see [stp:exceptional, §4.5, §10.1]. Also note that the definition implies that $F$ is totally real. That said, the “odd” case does have implications in certain CM settings. For example, let $F$ be quadratic imaginary, and let $\bar{\rho}\colon\Gamma_{F}\to\mathrm{GL}_{n}(\overline{\mathbb{F}}_{p})$ be an irreducible representation such that

[TABLE]

where $\mu\colon\Gamma_{\mathbb{Q}}\to\overline{\mathbb{F}}_{p}^{\times}$ is a character. Moreover assume that when we realize this essential conjugate self-duality as a relation

[TABLE]

for some $A\in\mathrm{GL}_{n}(\overline{\mathbb{F}}_{p})$ (and all $g\in\Gamma_{F}$), the scalar $A\cdot{}^{t}A^{-1}$ (which is easily seen to be $\pm 1$) actually equals $+1$. Then the pair $(\bar{\rho},\mu)$ can be extended to a homomorphism

[TABLE]

where $j^{2}=1$ and $j(g,a)j^{-1}=(a\cdot{}^{t}g^{-1},a)$, and this $\bar{r}$ is odd in the sense of Definition 1.2.

There are essentially two techniques for approaching cases of Question 1.1. For classical groups, automorphy lifting and potential automorphy theorems, via a technique introduced in [khare-wintenberger:serre0], yield the most robust results. For instance, the strongest lifting results in the previous example ($\bar{\rho}$ essentially conjugate self-dual over a quadratic imaginary field) follow from the work of Barnet-Lamb, Gee, Geraghty, and Taylor ([blggt:potaut]). For general $G$, however, we do not have a good understanding of automorphic Galois representations, and we must rely on purely Galois-theoretic methods. Ramakrishna developed the first such method in the papers [ramakrishna:lifting] and [ramakrishna02], which, as noted above, resolved Question 1.1 in the setting of Serre’s original modularity conjecture ($F=\mathbb{Q}$, $G=\mathrm{GL}_{2}$, $\bar{\rho}$ odd and irreducible). Our work relies on the methods of [ramakrishna:lifting] and [ramakrishna02], particularly as extended in the “doubling method” of [klr] and the work of Hamblen and Ramakrishna ([ramakrishna-hamblen]).

From now on, we will replace $\overline{\mathbb{Z}}_{p}$ (resp. $\overline{\mathbb{F}}_{p}$) by the ring of integers $\mathcal{O}$ in some finite extension $E$ of $\mathbb{Q}_{p}$ (resp. the residue field $k$ of $\mathcal{O}$). We let $\varpi$ denote a uniformizer of $\mathcal{O}$ and ${m_{\mathcal{O}}}=(\varpi)$ the maximal ideal of $\mathcal{O}$. Thus we now take $G$ as before but defined over $\mathcal{O}$, and we study continuous homomorphisms $\bar{\rho}\colon\Gamma_{F,{\mathcal{S}}}\to G(k)$, where ${\mathcal{S}}$ is a finite set of primes containing those above $p$, and $\Gamma_{F,{\mathcal{S}}}$ denotes $\mathrm{Gal}(F({\mathcal{S}})/F)$ for $F({\mathcal{S}})$ the maximal (inside $\overline{F}$) Galois extension of $F$ that is unramified away from ${\mathcal{S}}$.

There are several difficulties in extending the method of [ramakrishna02] to lifting odd irreducible representations to $G(k)$ for general groups:

• •

In the arguments of [ramakrishna02] one must construct at all primes $v$ at which $\bar{\rho}$ is ramified a formally smooth irreducible component of the local lifting ring $R^{\square}_{\bar{\rho}|_{\Gamma_{F_{v}}}}$ (for $v$ not above $p$) or a formally smooth component of the lifting ring that parametrizes lifts of a fixed inertial type and (Hodge–Tate regular) $p$-adic Hodge type (for $v$ above $p$). Such components do not always exist in the level of generality in which we work.

• •

The lack of smooth components as above, and more precisely the fact that $\bar{\rho}|_{\Gamma_{F_{v}}}$ may not have a Witt-vector valued lift corresponding to a formally smooth point of the generic fiber local lifting ring, necessitates working with more general (typically ramified) coefficients $\mathcal{O}$, while in *loc. cit. *one can work with the ring of Witt vectors $W(k)$. This causes complications related to the fact that if $\mathcal{O}$ is ramified, $\mathcal{O}/\varpi^{2}$ is of characteristic $p$ and hence isomorphic to the dual numbers $k\oplus k[\varepsilon]$.

• •

The auxiliary prime arguments of [ramakrishna02] break down as the image of $\bar{\rho}$ gets smaller. For general $G$, where many possible images can still lead to “irreducible” $\bar{\rho}$, this is a basic difficulty.

These difficulties are not as serious an impediment for $G=\mathrm{GL}_{2}$ as compared to the case of general $G$. In [ramakrishna02], under mild hypotheses on $\bar{\rho}|_{\Gamma_{\mathbb{Q}_{p}}}$, the necessary local theory is worked out (we should note, however, that particularly at the prime $p$ the situation is here considerably simplified by working over $\mathbb{Q}_{p}$ rather than a ramified extension). As for the global hypotheses, by a theorem of Dickson any irreducible subgroup of $\mathrm{GL}_{2}(k)$ (for $p\geq 7$) either has order prime to $p$, in which case one can take the “Teichmüller” lift, or has projective image conjugate to a subgroup of the form $\mathrm{PSL}_{2}(k^{\prime})$ or $\mathrm{PGL}_{2}(k^{\prime})$ for some subfield $k^{\prime}$ of $k$. This allows Ramakrishna to restrict to the case where the adjoint representation $\mathrm{ad}^{0}(\bar{\rho})$ is absolutely irreducible.

For higher-rank $G$, the global arguments of [ramakrishna02] work with little change under the corresponding assumption that the adjoint representation $\bar{\rho}(\mathfrak{g}^{\mathrm{der}})$ (this will be our notation for the Galois module $\mathfrak{g}^{\mathrm{der}}$, equipped with the action of $\Gamma_{F}$ via $\operatorname{Ad}\circ\bar{\rho}$) is absolutely irreducible. Such a generalization is carried out in [stp:exceptional]. The same paper also proves a variant with somewhat smaller image, in which $\operatorname{im}(\bar{\rho})$ contains (approximately) $\varphi(\mathrm{SL}_{2}(k))$, where $\varphi\colon\mathrm{SL}_{2}\to G$ is a principal $\mathrm{SL}_{2}$. In this case $\bar{\rho}(\mathfrak{g}^{\mathrm{der}})$ decomposes into $r$ irreducible factors, where $r$ is the semisimple rank of $G$, and the final result depended on an explicit analysis of this decomposition, requiring case-by-case calculations depending on the Dynkin type, with the result only verified for the exceptional groups via a computer calculation. More seriously, the method did not apply to groups of type $D_{2m}$, for which $\mathfrak{g}^{\mathrm{der}}$ is not multiplicity-free as an $\mathrm{SL}_{2}$-module (one factor occurs with multiplicity two). Some other instructive examples of how variants of the familiar Ramakrishna arguments still fail to treat relatively simple images can be found in [tang:thesisANT].

1.2. Main theorem

Before explaining how we overcome the difficulties mentioned above, we will state the main theorem. From now on we will require of $G$ that the component group $\pi_{0}(G)$ is finite étale of order prime to $p$. (See §LABEL:notation and the beginning of §LABEL:defprelimsection for the group theory and deformation theory notations.)

Theorem A** (See Theorem LABEL:mainthm).**

Let $p\gg_{G}0$ be a prime. Let $F$ be a totally real field, and let $\bar{\rho}\colon\Gamma_{F,{\mathcal{S}}}\to G(k)$ be a continuous representation unramified outside a finite set of finite places ${\mathcal{S}}$ containing the places above $p$. Let $\widetilde{F}$ denote the smallest extension of $F$ such that $\bar{\rho}(\Gamma_{\widetilde{F}})$ is contained in $G^{0}(k)$, and assume that $[\widetilde{F}(\zeta_{p}):\widetilde{F}]$ is strictly greater than the integer $a_{G}$ arising in Lemma LABEL:cyclicq (which depends only on the root datum of $G$). Fix a geometric lift $\mu\colon\Gamma_{F,{\mathcal{S}}}\to G/G^{\mathrm{der}}(\mathcal{O})$ of $\bar{\mu}:=\bar{\rho}\pmod{G^{\mathrm{der}}}$, and assume that $\bar{\rho}$ satisfies the following:

• •

$\bar{\rho}$* is odd, i.e. for all infinite places $v$ of $F$, $h^{0}(\Gamma_{F_{v}},\bar{\rho}(\mathfrak{g}^{\mathrm{der}}))=\dim(\mathrm{Flag}_{G^{\mathrm{der}}})$.*

• •

$\bar{\rho}|_{\Gamma_{\widetilde{F}(\zeta_{p})}}$* is absolutely irreducible.*

• •

For all $v\in{\mathcal{S}}$, $\bar{\rho}|_{\Gamma_{F_{v}}}$ has a lift $\rho_{v}\colon\Gamma_{F_{v}}\to G(\mathcal{O})$ of type $\mu|_{\Gamma_{F_{v}}}$; and that for $v\mid p$ this lift may be chosen to be de Rham and regular in the sense that the associated Hodge–Tate cocharacters are regular.

Then there exist a finite extension $E^{\prime}$ of $E=\operatorname{Frac}(\mathcal{O})$ (whose ring of integers and residue field we denote by $\mathcal{O}^{\prime}$ and $k^{\prime}$) depending only on the set $\{\rho_{v}\}_{v\in{\mathcal{S}}}$; a finite set of places $\widetilde{{\mathcal{S}}}$ containing ${\mathcal{S}}$; and a geometric lift

[TABLE]

of $\bar{\rho}$, and having projection to $G/G^{\mathrm{der}}(\mathcal{O^{\prime}})$ equal to $\mu$, such that $\rho(\Gamma_{F,\widetilde{{\mathcal{S}}}})$ contains $\widehat{G^{\mathrm{der}}}(\mathcal{O}^{\prime})$. Moreover, if we fix an integer $t_{0}$ and for each $v\in{\mathcal{S}}$ an irreducible component defined over $\mathcal{O}$ and containing $\rho_{v}$ of:

• •

for $v\in{\mathcal{S}}\setminus\{v\mid p\}$, the generic fiber of the local lifting ring, $R^{\square,\mu}_{\bar{\rho}|_{\Gamma_{F_{v}}}}[\frac{1}{\varpi}]$ (where $R^{\square,\mu}_{\bar{\rho}|_{\Gamma_{F_{v}}}}$ pro-represents $\operatorname{\mathrm{Lift}}_{\bar{\rho}|_{\Gamma_{F_{v}}}}^{\mu}$); and

• •

for $v\mid p$, the lifting ring $R_{\bar{\rho}|_{\Gamma_{F_{v}}}}^{\square,\mu,\tau,\mathbf{v}}[1/\varpi]$ whose $\overline{E}$-points parametrize lifts of $\bar{\rho}|_{\Gamma_{F_{v}}}$ with specified inertial type $\tau$ and Hodge type $\mathbf{v}$ (see [balaji, Prop. 3.0.12] for the construction of this ring);

*then *