A 2-adic automorphy lifting theorem for unitary groups over CM fields

Abstract

We prove a ‘minimal’ type automorphy lifting theorem for 2-adic Galois representations of unitary type, over imaginary CM fields. We use this to improve an automorphy lifting theorem of Kisin for ${{\mathrm{GL}}}_2$.

1 Introduction

In this paper, we study the deformation theory of 2-adic Galois representations of unitary type, over CM number fields. More precisely, let F be an imaginary CM number field with maximal totally real subfield $F^+$, and let $c \in {{\mathrm{Gal}}}(F{/}F^+)$ denote the non-trivial element. In this introduction, we say that a continuous representation

$$\begin{aligned} \rho {:}\,{{\mathrm{Gal}}}(\overline{F}{/}F) \rightarrow {{\mathrm{GL}}}_n(\overline{{\mathbb {Q}}}_p) \end{aligned}$$

is of unitary type if there is an isomorphism $\rho ^c \cong \rho ^\vee \otimes \epsilon ^{1-n}$, where $\epsilon$ denotes the p-adic cyclotomic character. The deformation theory of such representations in the case that p is odd has been studied in a series of papers beginning with [8], and culminating in the paper [5], where the authors prove very general automorphy and potential automorphy theorems for such Galois representations.

In contrast, there has been no study to date of the deformation theory of representations with 2-adic coefficients, except when $n = 2$ (in which case, it is essentially equivalent to consider representations $\rho {:}\,G_{F^+} \rightarrow {{\mathrm{GL}}}_2(\overline{{\mathbb {Q}}}_p)$ with no additional self-duality condition). In this connection, we mention the papers of Dickinson [10] (who proves a modularity lifting theorem for 2-adic representations under a supplementary local hypothesis at infinity) and Khare and Wintenberger [21] and Kisin [18] (who prove modularity lifting theorems for 2-adic representations without such a hypothesis). The additional difficulties that arise in the 2-adic case are of two main types: first, to get a good handle on the local lifting rings, especially at places dividing 2, and second, to get a good control of the relevant Galois cohomology groups.

In this paper, we prove an automorphy lifting theorem for 2-adic Galois representations of unitary type in arbitrary dimension:

Theorem 1.1

(Theorem 5.1) Let $n \ge 2$. Let F be an imaginary CM number field with totally real subfield $F^+$. Fix a prime p and an isomorphism $\iota {:}\,\overline{{\mathbb {Q}}}_p \cong {\mathbb {C}}$, and consider a continuous representation

$$\begin{aligned} \rho {:}\,G_F \rightarrow {{\mathrm{GL}}}_n(\overline{{\mathbb {Q}}}_p). \end{aligned}$$

Suppose that $\rho$ satisfies the following conditions:

• (i) There is an isomorphism $\rho ^c \cong \rho ^\vee \epsilon ^{1-n}$.

• (ii) The group $\overline{\rho }(G_{F(\zeta _p)}) \subset {{\mathrm{GL}}}_n(\overline{{\mathbb {F}}}_p)$ is adequate, in the sense of Definition 2.20.

• (iii) The representation $\rho$ is almost everywhere unramified.

• (iv) There exists a RACSDC automorphic representation $\pi$ of ${{\mathrm{GL}}}_n({\mathbb {A}}_F)$ such that:

  • (a) There is an isomorphism $\overline{r_\iota (\pi )} \cong \overline{\rho }$.

  • (b) For each finite place v of F, we have $r_\iota (\pi )|_{G_{F_v}} \sim \rho |_{G_{F_v}}$ (this condition is automatic if $\pi _v$ and $\rho |_{G_{F_v}}$ are both unramified). In particular, if v | p, then $\rho |_{G_{F_v}}$ and $r_\iota (\pi )|_{G_{F_v}}$ are potentially crystalline.

• (v) If $p = 2$ and n is even, then there exists a place $v | \infty$ of $F^+$ at which the pair $(\overline{\rho }, \epsilon ^{1-n})$ is strongly residually odd, in the sense of Definition 3.3.

Then $\rho$ is automorphic: there exists a RACSDC automorphic representation $\Pi$ of ${{\mathrm{GL}}}_n({\mathbb {A}}_F)$ such that $\rho \cong r_\iota (\Pi )$.

This is a theorem of ‘minimal type’. We can phrase its conclusion more colloquially as follows: given two n-dimensional Galois representations $\rho _1, \rho _2$ of unitary type, which have ‘the same’ local behaviour at each place v of F and the same residual representation, the automorphy of one implies the automorphy of the other. We state the theorem with no restriction on p since even in the case where p is odd, we are able to make a slight improvement on existing results (cf. [32, Theorem 7.1]).

The main ingredients in the proof of Theorem 1.1 which are new to this paper are the definition of a representable deformation functor in the case $p = 2$, and the observation that one can carry out the procedure of killing the dual Selmer group, even in the case where the base field F contains $p\text {th}$ roots of unity (see Proposition 2.21). It is here that we require the local hypothesis (v) at infinity. In the case $n = 2$, this condition is more-or-less equivalent to asking that for a given residual representation $\overline{\rho }{:}\,G_{F^+} \rightarrow {{\mathrm{GL}}}_2(\overline{{\mathbb {F}}}_p)$, there exist a place $v | \infty$ such that the image $\overline{\rho }(c_v)$ of complex conjugation at v is non-trivial. This is exactly the condition imposed by Dickinson [10].

The definition of ‘adequate subgroup’ that we use was first written down in the case $p = 2$ by Guralnick et al. [13]; it is pleasant to see that this turns out to be the right definition to prove automorphy lifting theorems in this case.

Having proved Theorem 1.1, we apply it to improve the aforementioned modularity lifting theorem of Kisin for ${{\mathrm{GL}}}_2$. We are able to prove the following result.

Theorem 1.2

(Theorem 6.1) Let F be a totally real number field, and let $\rho {:}\,G_F \rightarrow {{\mathrm{GL}}}_2(\overline{{\mathbb {Q}}}_2)$ be a continuous representation satisfying the following conditions:

• (i) $\rho$ is almost everywhere unramified.

• (ii) For each place v|2 of $F,\,\rho |_{G_{F_v}}$ is potentially crystalline. For each embedding $\tau {:}\,F_v \hookrightarrow \overline{{\mathbb {Q}}}_2,\,\mathrm {HT}_\tau (\rho ) = \{ 0, 1 \}$.

• (iii) $\overline{\rho }$ is absolutely irreducible, and has non-soluble image. There exists a place $v|\infty$ of F such that $\overline{\rho }(c_v)$ is non-trivial.

• (iv) There exists a RAESDC automorphic representation $(\pi , \chi )$ of ${{\mathrm{GL}}}_2({\mathbb {A}}_F)$ and an isomorphism $\iota {:}\,\overline{{\mathbb {Q}}}_2 \cong {\mathbb {C}}$ such that $\overline{r_\iota (\pi )} \cong \overline{\rho }$.

Then $\rho$ is automorphic: there exists a RAESDC automorphic representation $(\sigma , \psi )$ of ${{\mathrm{GL}}}_2({\mathbb {A}}_F)$ such that $\rho \cong r_\iota (\sigma )$.

An analogous improvement in the case $p > 2$ has been made by Barnet-Lamb et al. [3]. As in that work, the main problem is to construct ordinary automorphic lifts of a given (automorphic) residual representation, and we accomplish this by using Theorem 1.1 in the case of $n = 4$, by tensoring together various 2-dimensional representations with given local properties. Our techniques are broadly similar to those of [3], the main wrinkle being that induction from index 2 subgroups does not preserve adequacy. Where the authors of [3] use automorphic induction from quadratic extensions, we must therefore use tensor product functoriality for ${{\mathrm{GL}}}_2 \times {{\mathrm{GL}}}_2$, and tensor together residual representations which have insoluble image.

We observe that Theorem 1.2 is not strictly stronger than the theorems of [18]; rather, we have exchanged a condition at p for an apparently milder condition at infinity. This condition at infinity could be removed if the analogous hypothesis (v) of Theorem 1.1 could be removed. It is possible that this could be done using the techniques of [18, 21], although we have not tried to do this here. Given the important role played by 2-adic automorphy lifting theorems in the proof of Serre’s conjecture [20, 21], this seems like an interesting problem.

We now describe the organization of this paper. In Sect. 2, we define our deformation problem for Galois representations of unitary type, and study it using Galois cohomology. Where possible, we have given arguments that are independent of the residual characteristic p, although at some points it is impossible to avoid splitting up into cases (according as to whether p is even or odd). In Sect. 4, we define spaces of algebraic modular forms on definite unitary groups, and prove an automorphy lifting result under some additional local hypotheses. Given the foundations built up in Sect. 2, this section contains little that is new compared to [32, § 6], and we only sketch parts of the argument that remain unchanged. In Sect. 5, we prove Theorem 1.1, by reduction to the main result of Sect. 2. In Sect. 6, we apply this result to the proof of Theorem 1.2.

1.1 Notation

If E / F is a quadratic field extension, then we write $\delta _{E{/}F}{:}\,{{\mathrm{Gal}}}(E{/}F) \rightarrow \{ \pm 1 \}$ for the unique non-trivial character. A base number field F having been fixed, we will also choose algebraic closures $\overline{F}$ of F and $\overline{F}_v$ of $F_v$ for every finite place v of F. If p is a prime, then we will write $\overline{{\mathbb {Q}}}_p$ for a fixed choice of algebraic closure of ${\mathbb {Q}}_p$, and ${{\mathrm{val}}}_p$ for the p-adic valuation on ${\overline{{\mathbb {Q}}}_p}$ normalized so that ${{\mathrm{val}}}_p(p) = 1$. These choices define the absolute Galois groups $G_F = {{\mathrm{Gal}}}(\overline{F}{/}F)$ and $G_{F_v} = {{\mathrm{Gal}}}(\overline{F}_v{/}F_v)$. We write $I_{F_v} \subset G_{F_v}$ for the inertia subgroup. We also fix embeddings $\overline{F} \hookrightarrow \overline{F}_v$, extending the canonical embeddings $F \hookrightarrow F_v$. This determines for each place v of F an embedding $G_{F_v} \rightarrow G_{F}$. We write ${\mathbb {A}}_F$ for the adele ring of F, and ${\mathbb {A}}_F^\infty = \prod '_{v \not \mid \infty } F_v$ for its finite part. If v is a finite place of F, then we write k(v) for the residue field at v and $q_v = \# k(v)$.

We write ${{\mathrm{Frob}}}_v \in G_{F_v}{/}I_{F_v}$ for the geometric Frobenius element. We write $\epsilon {:}\,G_F \rightarrow {\mathbb {Z}}_p^\times$ for the p-adic cyclotomic character; if v is a finite place of F, not dividing p, then $\epsilon ({{\mathrm{Frob}}}_v) = q_v^{-1}$. If $\rho {:}\,G_F \rightarrow {{\mathrm{GL}}}_n({\overline{{\mathbb {Q}}}_p})$ is a continuous representation, we say that $\rho$ is de Rham if for each place v | p of $F,\,\rho |_{G_{F_v}}$ is de Rham. In this case, we can associate to each embedding $\tau {:}\,F \hookrightarrow {\overline{{\mathbb {Q}}}_p}$ a multiset $\mathrm {HT}_\tau (\rho )$ of Hodge–Tate weights, which depends only on $\rho |_{G_{F_v}}$, where v is the place of F induced by $\tau$. This multiset has n elements, counted with multiplicity. There are two natural normalizations for $\mathrm {HT}_\tau (\rho )$ which differ by a sign, and we choose the one with $\mathrm {HT}_\tau (\epsilon ) = \{ -1 \}$ for every choice of $\tau$.

We use geometric conventions for the Galois representations associated to automorphic forms. First, we use the normalizations of the local and global Artin maps ${{\mathrm{Art}}}_{F_v}{:}\,F_v^\times \rightarrow W_{F_v}^\text {ab}$ and ${{\mathrm{Art}}}_F{:}\,{\mathbb {A}}_F^\times \rightarrow G_F^\text {ab}$ which send uniformizers to geometric Frobenius elements. If $n \ge 1$ and v is a place of F, then we write ${{\mathrm{rec}}}_{F_v}$ for the local Langlands correspondence for ${{\mathrm{GL}}}_n(F_v)$, normalized as in [9, § 2.1]. If v is a finite place of F, then we define ${{\mathrm{rec}}}_{F_v}^T(\pi ) = {{\mathrm{rec}}}_{F_v}(\pi \otimes | \cdot |^{(1-n){/}2})$. Then ${{\mathrm{rec}}}_{F_v}^T$ commutes with automorphisms of ${\mathbb {C}}$, and so makes sense over any field $\Omega$ which is abstractly isomorphic to ${\mathbb {C}}$ (e.g. ${\overline{{\mathbb {Q}}}_p}$). We write $\Vert \cdot \Vert {:}\,{\mathbb {A}}_F^\times \rightarrow {\mathbb {R}}_{>0}$ for the standard norm character, which corresponds under global class field theory to the cyclotomic character $\epsilon$.

If (r, N) is any Weil–Deligne representation, we write $(r, N)^\text {F-ss}$ for its Frobenius–semi-simplification. If v is a finite place of F and $\rho {:}\,G_{F_v} \rightarrow {{\mathrm{GL}}}_n({\overline{{\mathbb {Q}}}_p})$ is a continuous representation, which is de Rham if v | p, then we write $\mathrm {WD}(\rho )$ for the associated Weil–Deligne representation, which is uniquely determined by $\rho$, up to isomorphism.

We will call a finite extension $E{/}{\mathbb {Q}}_p$ inside ${\overline{{\mathbb {Q}}}_p}$ a coefficient field. A coefficient field E having been fixed, we will write ${\mathcal {O}}$ or ${\mathcal {O}}_E$ for its ring of integers, k or $k_E$ for its residue field, and $\lambda$ or $\lambda _E$ for its maximal ideal. If A is a complete Noetherian local ${\mathcal {O}}$-algebra with residue field k, then we write ${\mathfrak {m}}_A \subset A$ for its maximal ideal, and ${\mathrm {CNL}}_A$ for the category of complete Noetherian local A-algebras with residue field k. We endow each object $R \in {\mathrm {CNL}}_A$ with its profinite (${\mathfrak {m}}_R$-adic) topology.

If $\Gamma$ is a profinite group and $\rho {:}\,\Gamma \rightarrow {{\mathrm{GL}}}_n({\overline{{\mathbb {Q}}}_p})$ is a continuous representation, then we can assume (after a change of basis) that $\rho$ takes values in ${{\mathrm{GL}}}_n({\mathcal {O}})$, for some choice of coefficient field E. The semi-simplification of the composite representation $\Gamma \rightarrow {{\mathrm{GL}}}_n({\mathcal {O}}) \rightarrow {{\mathrm{GL}}}_n(k)$ is independent of choices, up to isomorphism, and we will write $\overline{\rho }{:}\,\Gamma \rightarrow {{\mathrm{GL}}}_n({\overline{{\mathbb {F}}}_p})$ for this semi-simplification.

If E is a coefficient field and $\overline{\rho }{:}\,\Gamma \rightarrow {{\mathrm{GL}}}_n(k)$ is a continuous representation, then we write ${{\mathrm{ad}}}\overline{\rho }$ for ${{\mathrm{End}}}_k(\overline{\rho })$, endowed with its structure of $k[\Gamma ]$-module. We write ${{\mathrm{ad}}}^0 \overline{\rho } \subset {{\mathrm{ad}}}\overline{\rho }$ for the submodule of trace 0 endomorphisms, and (if $\Gamma = G_F$ for a number field F) ${{\mathrm{ad}}}^0 \overline{\rho }(1)$ for its twist by the cyclotomic character. We write ${{\mathrm{ad}}}_0 \overline{\rho }$ for the quotient of ${{\mathrm{ad}}}\overline{\rho }$ by the $\Gamma$-invariant subspace of scalar endomorphisms. If M is a discrete ${\mathbb {Z}}[G_F]$-module, then we write $H^1(F, M)$ for the continuous Galois cohomology group with coefficients in M. Similarly, if M is a discrete ${\mathbb {Z}}[G_{F_v}]$-module, then we write $H^1(F_v, M)$ for the continuous Galois cohomology group with coefficients in M. If M is a discrete $k[G_F]$-module (resp. $k[G_{F_v}]$-module), then $H^1(F, M)$ (resp. $H^1(F_v, M)$) is a k-vector space, and we write $h^1(F, M)$ (resp. $h^1(F_v, M)$) for the dimension of this k-vector space, provided that it is finite.

2 Deformation theory

Let $n \ge 1$ be an integer, and define ${\mathcal {G}}_n = ({{\mathrm{GL}}}_n \times {{\mathrm{GL}}}_1) \rtimes \{ 1, \jmath \}$, an algebraic group over ${\mathbb {Z}}$. The semi-direct product is defined by the relation

$$\begin{aligned} \jmath (g, \mu ) = (\mu {}^t g^{-1}, \mu ) \jmath . \end{aligned}$$

We let ${\mathfrak {g}}_n = {{\mathrm{Lie}}}{\mathcal {G}}_n$. There is a character $\nu {:}\,{\mathcal {G}}_n \rightarrow {\mathbb {G}}_m$ given by the formula $(g, \mu ) \mapsto \mu ,\,\jmath \mapsto -1$. Let $\Gamma$ be a group, and let $\Delta$ be an index 2 subgroup. The reason for introducing the group ${\mathcal {G}}_n$ is the following lemma ([8, Lemma 2.1.1]):

Lemma 2.1

Suppose that R is a ring, and let $c_0 \in \Gamma - \Delta$. Then the following two sets are in natural bijection:

• (i) The set of homomorphisms $r{:}\,\Gamma \rightarrow {\mathcal {G}}_n(R)$ such that $r^{-1}({\mathcal {G}}_n^0(R)) = \Delta$.

• (ii) The set of triples $(\rho , \mu , \langle \cdot , \cdot \rangle )$, where $\rho {:}\,\Delta \rightarrow {{\mathrm{GL}}}_n(R)$ and $\mu {:}\,\Gamma \rightarrow R^\times$ are homomorphisms and $\langle \cdot , \cdot \rangle$ is a perfect R-linear pairing on $R^n$ such that for all $x, y \in R^n,\,\delta \in \Delta$, we have

  $$\begin{aligned} \langle x, \rho (c_0^2) y \rangle = - \mu (c_0)\langle y, x \rangle \text { and }\left\langle \rho (\delta )x, \rho (\delta ^{c_0})y \right\rangle = \mu (\delta ) \langle x, y \rangle . \end{aligned}$$

Under this correspondence we have $\mu (\gamma ) = (\nu \circ r)(\gamma )$ for all $\gamma \in \Gamma$, and $\langle x, y \rangle = {}^t x A^{-1} y$, where $r(c_0) = (A, -\mu (c_0))\jmath$. If $\Gamma$ and R are topological groups and $\Delta \subset \Gamma$ is a closed subgroup, then continuous homomorphisms r correspond to pairs of continuous homomorphisms $(\rho , \mu )$.

The group ${\mathcal {G}}_n^0(R) = {{\mathrm{GL}}}_n(R) \times {{\mathrm{GL}}}_1(R)$ acts by conjugation on the set of homomorphisms $r{:}\,\Gamma \rightarrow {\mathcal {G}}_n(R)$ such that $r^{-1}({\mathcal {G}}_n^0(R)) = \Delta$. We observe that, in general, the ${{\mathrm{GL}}}_1(R)$ factor does not act trivially; rather, under the dictionary of Lemma 2.1, it acts by rescaling the perfect pairing $\langle \cdot , \cdot \rangle$.

If $r{:}\,\Gamma \rightarrow {\mathcal {G}}_n(R)$ is a homomorphism such that $\Delta = r^{-1}({\mathcal {G}}_n^0(R))$, then we will write $r|_\Delta {:}\,\Delta \rightarrow {{\mathrm{GL}}}_n(R)$ for the representation that arises by restricting r to $\Delta$ and then projecting to the ${{\mathrm{GL}}}_n$ factor of ${\mathcal {G}}_n^0 = {{\mathrm{GL}}}_n \times {{\mathrm{GL}}}_1$. Thus it makes sense to speak, for example, of the characteristic polynomial of an element $r|_\Delta (\delta ),\,\delta \in \Delta$. We write ${\mathfrak {g}}_n r$ for the $R[\Gamma ]$-module, free as R-module, induced by the adjoint representation of ${\mathcal {G}}_n$.

Lemma 2.2

Let k be a field, let $\rho {:}\,\Delta \rightarrow {{\mathrm{GL}}}_n(k)$ be an absolutely irreducible homomorphism, and let $\mu {:}\,\Gamma \rightarrow k^\times$ be a character such that $\rho ^{c_0} \cong \rho ^\vee \otimes \mu$. Then there exists a homomorphism $r{:}\,\Gamma \rightarrow {\mathcal {G}}_n(k)$ such that $r|_\Delta = \rho ,\,\nu \circ r |_\Delta = \mu$, and $r(c_0) \in {\mathcal {G}}_n(k) - {\mathcal {G}}_n^0(k)$. This extension is unique up to ${\mathcal {G}}^0_n(k)$-conjugacy.

Proof

By hypothesis, there exists a perfect pairing $\langle \cdot , \cdot \rangle {:}\,k^n \times k^n \rightarrow k$ such that for all $\delta \in \Delta ,\,x, y \in k^n,\,\langle \rho (\delta ) x, \rho (\delta ^{c_0}) y \rangle = \mu (\delta ) \langle x, y \rangle$. This pairing is unique up to $k^\times$-multiple (because $\rho$ is absolutely irreducible). Define a new pairing by the formula $\langle x, y \rangle ' = \langle x, \rho (c_0^2) y \rangle$. Then a calculation shows $\langle \rho (\delta ) x, \rho (\delta ^{c_0}) y \rangle ' = \mu (\delta ^{c_0}) \langle x, y \rangle ' = \mu (\delta ) \langle x, y \rangle '$, hence $\langle x, y \rangle ' = \alpha \langle x, y \rangle$ for some $\alpha \in k^\times$.

Define another new pairing by $\langle x, y \rangle '' = \langle y, \rho (c_0^2) x \rangle '$. Then another calculation shows that

$$\begin{aligned} \langle x, y \rangle '' = \mu (c_0)^2 \langle x, y \rangle = \alpha ^2 \langle x, y \rangle , \end{aligned}$$

hence $\alpha = \pm \mu (c_0)$. After possibly replacing $\mu$ by its multiple by the non-trivial character of $\Gamma {{/}}\Delta$, we see that the triple $(\rho , \mu , \langle \cdot , \cdot \rangle )$ is of the type appearing in the statement of Lemma 2.1, and the existence of r follows from this.

If $r'$ is another such extension, then it corresponds under the dictionary of Lemma 2.1 to a triple $(\rho , \mu , \lambda \langle \cdot , \cdot \rangle )$ for some $\lambda \in k^\times$. We thus have $r' = (1, \lambda ) r (1, \lambda )^{-1}$, showing that r and $r'$ are indeed ${\mathcal {G}}_n^0(k)$-conjugate. $\square$

The following lemma is [8, Lemma 2.1.5].

Lemma 2.3

Suppose that $\Gamma$ is a profinite group and $r{:}\,\Gamma \rightarrow {{\mathrm{GL}}}_n(\overline{{\mathbb {Q}}}_p)$ is a continuous representation with $\Delta = r^{-1}({\mathcal {G}}_n^0(\overline{{\mathbb {Q}}}_p))$. Then there exists a finite extension $E{/}{\mathbb {Q}}_p$ inside $\overline{{\mathbb {Q}}}_p$ and another continuous representation $r'{:}\,\Gamma \rightarrow {\mathcal {G}}_n({\mathcal {O}}_E)$ which is ${\mathcal {G}}_n^0(\overline{{\mathbb {Q}}}_p)$-conjugate to r.

Lemma 2.4

Suppose that $R \supset S$ are complete Noetherian local rings with common residue field k and ${\mathfrak {m}}_R \cap S = {\mathfrak {m}}_S$. Suppose that $\Gamma$ is a profinite group and $r{:}\,\Gamma \rightarrow {\mathcal {G}}_n(R)$ is a continuous representation with $\Delta = r^{-1}({\mathcal {G}}_n^0(R))$. Suppose finally that $r|_\Delta \text { mod } {\mathfrak {m}}_R$ is absolutely irreducible and that ${{\mathrm{tr}}}r|_\Delta (\Delta ) \subset S$ and $\nu \circ r$ is valued in $S^\times \subset R^\times$. Then there exists a continuous representation $r'{:}\,\Gamma \rightarrow {\mathcal {G}}_n(S)$ and $g\in \ker ({\mathcal {G}}_n(R) \rightarrow {\mathcal {G}}_n(k))$ such that $r' = g r g^{-1}$.

Proof

By the analogous result for ${{\mathrm{GL}}}_n$ (i.e. [8, Lemma 2.1.10]) and our assumption that $\nu \circ r$ is valued in $S^\times$, we can assume that $r(\Delta ) \subset {\mathcal {G}}_n^0(S) = {{\mathrm{GL}}}_n(S) \times {{\mathrm{GL}}}_1(S)$. Choose $c_0 \in \Gamma - \Delta$, and write $r(c_0) = (A, -\mu )\jmath$. Let $\rho = r|_\Delta$. Then we have $\rho ^{c_0} = A \rho ^\vee A^{-1} \otimes (\nu \circ r)|_\Delta$, and hence (by [8, Lemma 2.1.9]) we can find $B \in {{\mathrm{GL}}}_n(S)$ such that $\rho ^{c_0} = B \rho ^\vee B^{-1} \otimes (\nu \circ r)|_\Delta$.

By Schur’s lemma ([8, Lemma 2.1.8]), we have $A = \alpha B$ for some $\alpha \in R^\times$. Since R and S have the same residue field, we can assume (after multiplying B by an element of $S^\times$) that $\alpha \in 1 + {\mathfrak {m}}_R$. We then have

$$\begin{aligned} (1, \alpha ) r(c_0) (1, \alpha )^{-1} = (1, \alpha ) (A, -\mu )\jmath (1, \alpha )^{-1} = (\alpha ^{-1} A, -\mu ) \jmath \in {\mathcal {G}}_n(S). \end{aligned}$$

It follows that $r' = (1, \alpha ) r (1, \alpha ^{-1})$ is valued in ${\mathcal {G}}_n(S)$, as desired. $\square$

A similar argument proves:

Lemma 2.5

Suppose that R is a complete Noetherian local ring, that $\Gamma$ is a profinite group and that $r_1, r_2{:}\,\Gamma \rightarrow {\mathcal {G}}_n(R)$ are continuous representations with $\Delta = r_1^{-1}({\mathcal {G}}_n^0(R)) = r_2^{-1}({\mathcal {G}}_n^0(R)),\,r_1 \text { mod }{\mathfrak {m}}_R = r_2 \text { mod }{\mathfrak {m}}_R$, and $\nu \circ r_1 = \nu \circ r_2$. Suppose moreover that $r_1|_\Delta \text { mod } {\mathfrak {m}}_R$ is absolutely irreducible, and that for all $\delta \in \Delta ,\,{{\mathrm{tr}}}r_1|_\Delta (\delta ) = {{\mathrm{tr}}}r_2|_\Delta (\delta )$. Then there exists $g \in \ker ( {\mathcal {G}}_n(R) \rightarrow {\mathcal {G}}_n(k) )$ such that $g r_1 g^{-1} = r_2$.

2.1 Galois deformation theory

Let p be a prime, and let $E{{/}}{\mathbb {Q}}_p$ be a coefficient field with ring of integers ${\mathcal {O}}$, maximal ideal $\lambda$, and residue field $k = {\mathcal {O}}{/}\lambda$. If A is a group functor on the category of ${\mathcal {O}}$-algebras, then we define for $R \in {\mathrm {CNL}}_{\mathcal {O}}\widehat{A}(R) = \ker ( A(R) \rightarrow A(k))$. Fix an imaginary CM number field F with maximal totally real subfield $F^+$. We assume that the following conditions are in effect:

• The extension $F{/}F^+$ is everywhere unramified.

• Each place of $F^+$ dividing p is split in F.

Let $c \in G_{F^+}$ be a fixed choice of complex conjugation. Let $S_p$ denote the set of places of $F^+$ dividing $p,\,S_\infty$ the set of places dividing $\infty$, and let S be a finite set of places of $F^+$ which contains $S_p \cup S_\infty$. We write F(S) for the maximal Galois extension of F unramified outside S; then $F(S){/}F^+$ is Galois. We define $G_{F^+, S} = {{\mathrm{Gal}}}(F(S){/}F^+),\,G_{F, S} = {{\mathrm{Gal}}}(F(S){/}F)$. Then $G_{F, S} \subset G_{F^+, S}$ is an index 2 normal subgroup, and the quotient $G_{F^+, S}{/}G_{F, S}$ is generated by the image of c.

We fix a continuous representation $\overline{r}{:}\,G_{F^+, S} \rightarrow {\mathcal {G}}_n(k)$ such that $\overline{r}^{-1}({\mathcal {G}}_n^0(k)) = G_{F, S}$ and $\overline{r}|_{G_{F, S}}$ is absolutely irreducible. We also fix a continuous character $\chi {:}\,G_{F^+, S} \rightarrow {\mathcal {O}}^\times$ such that $\overline{\chi } = \nu \circ \overline{r}$.

We can now give the definitions relating to the local and global deformation theory of the representation $\overline{r}$.

Definition 2.6

Let $v \in S$. We define the functor $\mathrm {Lift}_{v}^\square {:}\,{\mathrm {CNL}}_{\mathcal {O}}\rightarrow \mathrm {Sets}$ of unrestricted liftings of $\overline{r}|_{G_{F^+_v}}$ as follows: if $R \in {\mathrm {CNL}}_{\mathcal {O}}$, then $\mathrm {Lift}_v^\square (R)$ is the set of homomorphisms $r_v{:}\,G_{F^+_v} \rightarrow {\mathcal {G}}_n(R)$ such that $\nu \circ r_v = \chi |_{G_{F^+_v}}$ and $r_v \text { mod }{\mathfrak {m}}_R = \overline{r}|_{G_{F_v^+}}$.

By definition, a local deformation problem is a representable subfunctor ${\mathcal {D}}_v \subset \mathrm {Lift}_v^\square$ such that for all $R \in {\mathrm {CNL}}_{\mathcal {O}}$, the subset ${\mathcal {D}}_v(R) \subset \mathrm {Lift}_v^\square$ is invariant under the conjugation action of the group $\widehat{{\mathcal {G}}}_n(R)$.

We remark that the above definition depends on the choice of character $\chi$, although we do not include this in the notation. It is easy to see that for $v \in S$, the functor $\mathrm {Lift}_v^\square$ is a local deformation problem (i.e. that $\mathrm {Lift}_v^\square$ is represented by an object $R_v^\square \in {\mathrm {CNL}}_{\mathcal {O}}$).

Definition 2.7

A global deformation problem is a tuple

$$\begin{aligned} {\mathcal {S}}= \left( F, \overline{r}, {\mathcal {O}}, \chi , S, \{ {\mathcal {D}}_v \}_{v \in S}\right) , \end{aligned}$$

where $F,\,\overline{r},\,{\mathcal {O}},\,\chi$ and S are as above and for each $v \in S,\,{\mathcal {D}}_v$ is a local deformation problem.

Let $R \in {\mathrm {CNL}}_{\mathcal {O}}$. A lifting of $\overline{r}$ to R of type ${\mathcal {S}}$ is, by definition, a homomorphism $r{:}\,G_{F^+, S} \rightarrow {\mathcal {G}}_n(R)$ satisfying the following conditions:

• (i) We have $r \text { mod } {\mathfrak {m}}_R = \overline{r}$ and $\nu \circ r = \chi$.

• (ii) For each $v \in S$, we have $r|_{G_{F^+_v}} \in {\mathcal {D}}_v(R)$.

Two liftings $r_1, r_2$ of type ${\mathcal {S}}$ are said to be strictly equivalent if there exists $g \in \widehat{{\mathcal {G}}}_n(R)$ such that $g r_1 g^{-1} = r_2$. A strict equivalence class of liftings of type ${\mathcal {S}}$ is called a deformation.

Lemma 2.8

Let ${\mathcal {S}}$ be a global deformation problem, and let $\mathrm {Def}_{\mathcal {S}}{:}\,{\mathrm {CNL}}_{\mathcal {O}}\rightarrow \mathrm {Sets}$ be the functor which assigns to each $R \in {\mathrm {CNL}}_{\mathcal {O}}$ the set of deformations of $\overline{r}$ to R of type ${\mathcal {S}}$. Then $\mathrm {Def}_{\mathcal {S}}$ is representable.

Proof

Let $\mathrm {Def}_{\mathcal {S}}^\square {:}\,{\mathrm {CNL}}_{\mathcal {O}}\rightarrow \mathrm {Sets}$ be the functor of liftings of $\overline{r}$ of type ${\mathcal {S}}$. It is easy to see that $\mathrm {Def}_{\mathcal {S}}^\square$ is representable. Let $i{:}\,\widehat{{{\mathrm{GL}}}}_1 \rightarrow \widehat{{\mathcal {G}}}_n = \widehat{{{\mathrm{GL}}}}_n \times \widehat{{{\mathrm{GL}}}}_1$ be the map $\lambda \mapsto (\lambda \cdot 1_n, \lambda ^2)$, and let $\widehat{{\mathcal {H}}}_n$ be the functor in groups on ${\mathrm {CNL}}_{\mathcal {O}}$ given by the formula $\widehat{{\mathcal {H}}}_n(R) = \widehat{{\mathcal {G}}}_n(R){/}i(\widehat{{{\mathrm{GL}}}}_1(R))$.

It follows from [21, Proposition 2.5] that the functor $\widehat{{\mathcal {H}}}_n$ is representable by an object of ${\mathrm {CNL}}_{\mathcal {O}}$, formally smooth over ${\mathcal {O}}$. Moreover, $\widehat{{\mathcal {H}}}_n$ acts freely on $\mathrm {{{\mathrm{Def}}}}^\square _{\mathcal {S}}$. Indeed, let $R \in {\mathrm {CNL}}_{\mathcal {O}},\,r \in {{\mathrm{Def}}}_{\mathcal {S}}^\square (R)$, and suppose that $g \in {{\mathrm{Stab}}}_{\widehat{{\mathcal {G}}}_n(R)}(r)$. It follows from [8, Lemma 2.1.8] and our assumption that $\overline{r}|_{G_{F, S}}$ is absolutely irreducible that we can write $g = (\lambda \cdot 1_n , \mu )$, with $\lambda , \mu \in R^\times$. It follows from the equality $g r(c) g^{-1} = r(c)$ that in fact $\mu = \lambda ^2$, hence $g \in i(\widehat{{{\mathrm{GL}}}}_1(R))$, hence the image of g in $\widehat{{\mathcal {H}}}_n(R)$ is trivial. The lemma now follows from another application of [21, Proposition 2.5]. $\square$

We write $R_{\mathcal {S}}$ for the representing object of $\mathrm {Def}_{\mathcal {S}}$.

Definition 2.9

Let ${\mathcal {S}}$ be a global deformation problem, and let $T \subset S$. We define a T-framed lifting of $\overline{r}$ of type ${\mathcal {S}}$ to be a tuple $(r, \{ \alpha _v \}_{v \in T})$, where r is a lifting of type ${\mathcal {S}}$ and for each $v \in T,\,\alpha _v \in \widehat{{\mathcal {G}}}_n(R)$. Two T-framed liftings $(r_1, \{ \alpha _v \}_{v \in T})$ and $(r_2, \{ \beta _v \}_{v \in T})$ are said to be strictly equivalent if there exists $g \in \widehat{{\mathcal {G}}}_n(R)$ such that $g r_1 g^{-1} = r_2$ and $g \alpha _v = \beta _v$ for each $v \in T$. A strict equivalence class of T-framed liftings is called a T-framed deformation.

Lemma 2.10

Let ${\mathcal {S}}$ be a global deformation problem, and let $\mathrm {Def}_{\mathcal {S}}^T{:}\,{\mathrm {CNL}}_{\mathcal {O}}\rightarrow \mathrm {Sets}$ be the functor which associates to each $R \in {\mathrm {CNL}}_{\mathcal {O}}$ the set of T-framed deformations of $\overline{r}$ to R of type ${\mathcal {S}}$. Then $\mathrm {Def}_{\mathcal {S}}^T$ is representable.

Proof

If $T = \emptyset$, then this is just Lemma 2.8. The general case follows easily from [21, Proposition 2.5]. $\square$

We write $R_{\mathcal {S}}^T$ for the representing object of $\mathrm {Def}_{\mathcal {S}}^T$.

Lemma 2.11

Let ${\mathcal {S}}$ be a global deformation problem and let $T \subset S$ be non-empty. Then $R_{\mathcal {S}}^T$ is a formally smooth $R_{\mathcal {S}}$-algebra of relative dimension $(n^2 + 1)\# T - 1$.

Proof

The map $R_{\mathcal {S}}\rightarrow R_{\mathcal {S}}^T$ arises from universality (forget the framing). The rest of the lemma is easy. $\square$

Let ${\mathcal {S}}$ be a global deformation problem, and let $T \subset S$. Then there is a natural transformation

$$\begin{aligned} {{\mathrm{Def}}}_{\mathcal {S}}^T \rightarrow \prod _{v \in T} {\mathcal {D}}_v, \end{aligned}$$

given by the formula

$$\begin{aligned} \left( r_1, \{ \alpha _v \}_{v \in T}\right) \mapsto \left( \alpha _v r_1|_{G_{F_v^+}} \alpha _v^{-1}\right) _{v \in T}. \end{aligned}$$

(it is easy to check that this is independent of the chosen representatives). If we write $R_v \in {\mathrm {CNL}}_{\mathcal {O}}$ for the representing object of ${\mathcal {D}}_v$ ($v \in T$), then there is a corresponding morphism in ${\mathrm {CNL}}_{\mathcal {O}}$:

$$\begin{aligned} R^\text {loc}_{{\mathcal {S}}, T} = \widehat{\otimes }_{v \in T} R_v \rightarrow R_{\mathcal {S}}^T. \end{aligned}$$

2.2 Galois cohomology calculations

We continue with the notation of the previous section, and fix furthermore a choice of global deformation problem

$$\begin{aligned} {\mathcal {S}}= \left( F, \overline{r}, {\mathcal {O}}, \chi , S, \{ {\mathcal {D}}_v \}_{v \in S}\right) . \end{aligned}$$

If $v \in S$, then the set $\mathrm {Lift}_v^\square (k[\epsilon ])$ of liftings of $\overline{r}|_{G_{F_v^+}}$ to $k[\epsilon ]$ can be identified with the group of cocycles $Z^1(F_v^+, {{\mathrm{ad}}}\overline{r})$, via the formula

$$\begin{aligned} r(\sigma ) = \left( 1 + \epsilon \phi (\sigma )\right) \overline{r}(\sigma ) \text { }\text { }\text { }\left( \phi \in Z^1\left( F_v^+, {{\mathrm{ad}}}\overline{r}\right) , \text { }\sigma \in G_{F^+_v}\right) . \end{aligned}$$

Here we identify ${{\mathrm{ad}}}\overline{r} = {{\mathrm{End}}}_k(\overline{r}) = \widehat{{{\mathrm{GL}}}}_n(k[\epsilon ])$. Two such liftings $r_1, r_2$ are conjugate under the action of $\widehat{{\mathcal {G}}}_n(k[\epsilon ])$ if and only if the associated cocycles $\phi _1, \phi _2$ have the same image in the group $H^1(F_v^+, {\mathfrak {g}}_n \overline{r})$. If v is split in F, then the exact sequence of $k[G_{F^+, S}]$-modules

$$\begin{aligned} 0 \rightarrow {{\mathrm{ad}}}\overline{r} \rightarrow {\mathfrak {g}}_n \overline{r} \rightarrow k \rightarrow 0 \end{aligned}$$

splits over $k[G_{F^+_v}]$, and this is equivalent to asking that $\phi _1, \phi _2$ have the same image in $H^1(F^+_v, {{\mathrm{ad}}}\overline{r})$. We write ${\mathcal {L}}^1_v \subset Z^1(F_v^+, {{\mathrm{ad}}}\overline{r})$ for the k-vector space of cocycles corresponding to liftings in ${\mathcal {D}}_v(k[\epsilon ]) \subset \mathrm {Lift}_v^\square (k[\epsilon ])$. We write ${\mathcal {L}}_v$ for the image of ${\mathcal {L}}^1_v$ in $H^1(F^+_v, {{\mathrm{ad}}}\overline{r})$, and $\ell ^1_v = \dim _k {\mathcal {L}}_v^1,\,\ell _v = \dim _k {\mathcal {L}}_v$.

Let $T \subset S$. We now want to define some global cohomology groups $H^i_{{\mathcal {S}}, T}$ which can be used to analyze the relative tangent space of the morphism

$$\begin{aligned} R^\text {loc}_{{\mathcal {S}}, T} = \widehat{\otimes }_{v \in T} R_v \rightarrow R_{\mathcal {S}}^T. \end{aligned}$$

To this end, we define a chain complex $C^i_{{\mathcal {S}}, T}$ by the following formulae:

$$\begin{aligned} C^i_{{\mathcal {S}}, T} = \left\{ \begin{array}{ll} 0 &{}\quad i < 0; \\ C^0(F(S){/}F^+, {\mathfrak {g}}_n \overline{r}) &{}\quad i = 0; \\ C^1(F(S){/}F^+, {{\mathrm{ad}}}\overline{r}) \oplus _{v \in T} C^0(F_v^+, {\mathfrak {g}}_n \overline{r}) &{}\quad i = 1;\\ C^2(F(S){/}F^+, {{\mathrm{ad}}}\overline{r}) \oplus _{v \in T} C^1(F_v^+, {{\mathrm{ad}}}\overline{r}) \oplus _{v \in S - T} C^1(F_v^+, {{\mathrm{ad}}}\overline{r}){/}{\mathcal {L}}_v^1 &{}\quad i = 2; \\ C^i(F(S){/}F^+, {{\mathrm{ad}}}\overline{r}) \oplus _{v \in S} C^{i-1}(F(S){/}F^+, {{\mathrm{ad}}}\overline{r}) &{}\quad i \ge 3. \end{array}\right. \end{aligned}$$

The differentials are given by the formula

$$\begin{aligned} \partial (\phi , (\psi _v)_{v \in S}) = (\partial \phi , (\phi |_v - \partial \psi _v)_{v \in S}). \end{aligned}$$

It is easy to check that this is indeed a complex, and that the cohomology groups $H^i_{{\mathcal {S}}, T}$ of this complex fit into a long exact sequence

(1)

(the superscript ${}^\eta$ indicates that we take the image of cohomology with coefficients in ${{\mathrm{ad}}}\overline{r}$ in cohomology with coefficients in ${\mathfrak {g}}_n \overline{r}$. A similar notation is used in [21, § 4.1.4]). We define $h^i_{{\mathcal {S}}, T} = \dim _k H^i_{{\mathcal {S}}, T}$.

Lemma 2.12

Let notation be as above. We have $\dim _k {\mathfrak {m}}_{R_{\mathcal {S}}^T}{/}({\mathfrak {m}}_{R_{{\mathcal {S}}, T}^\text {loc}}, {\mathfrak {m}}_{R_{\mathcal {S}}^T}^2) = h^1_{{\mathcal {S}}, T}$. Consequently, there is a surjection $R^\text {loc}_{{\mathcal {S}}, T}\llbracket X_1, \ldots , X_g \rrbracket \rightarrow R_{{\mathcal {S}}}^T$ of $R^\text {loc}_{{\mathcal {S}}, T}$-algebras with $g = h^1_{{\mathcal {S}}, T}$.

Proof

We observe that there is an isomorphism

$$\begin{aligned} {{\mathrm{Hom}}}_k\left( {\mathfrak {m}}_{R_{\mathcal {S}}^T}{{/}}({\mathfrak {m}}_{R_{{\mathcal {S}}, T}^\text {loc}}, {\mathfrak {m}}_{R_{\mathcal {S}}^T}^2), k\right) \cong {{\mathrm{Hom}}}\left( R_{\mathcal {S}}^T{/}{\mathfrak {m}}_{R^\text {loc}_{{\mathcal {S}}, T}}, k[\epsilon ]\right) , \end{aligned}$$

and the latter space corresponds to equivalence classes of tuples $(r, (\alpha _v)_{v \in T})$, where r is a lifting of $\overline{r}$ to $k[\epsilon ]$ of type ${\mathcal {S}},\,\alpha _v \in \widehat{{\mathcal {G}}}_n(k[\epsilon ])$, and for each $v \in T,\,\alpha _v^{-1} r|_{G_{F^+_v}} \alpha _v$ is the trivial lifting. We can write $r = (1 + \epsilon \phi ) \overline{r}$ and $\alpha _v = 1 + \epsilon _v a_v$ with $\phi \in Z^1(F_v^+, {{\mathrm{ad}}}\overline{r})$ and $a_v \in {\mathfrak {g}}_n \overline{r}$, and $\partial (\phi , (a_v)_{v \in T}) = 0$. The pairs $(\phi , (a_v)_{v\in T})$ and $(\phi ', (a'_v)_{v \in T})$ define equivalent T-framed deformations if and only if there exists $b \in {\mathfrak {g}}_n \overline{r}$ such that $\phi - \phi ' = \partial b$ and $a_v - a'_v = b$ for each $v \in T$. It is now easy to check that the space of such equivalence classes of cocycles is canonically identified with $H^1_{{\mathcal {S}}, T}$. $\square$

If M is any $k[G_{F^+, S}]$-module, then we define the Euler characteristics

$$\begin{aligned} \chi (F_v^+, M)= & {} \sum _{i=0}^2 (-1)^i h^i(F_v^+, M),\\ \chi (F(S){/}F^+, M)= & {} \sum _{i = 0}^2 (-1)^i h^i(F(S){/}F^+, M) \end{aligned}$$

and

$$\begin{aligned} \chi _{{\mathcal {S}}, T} = \sum _{i = 0}^3 (-1)^i h^i_{{\mathcal {S}}, T}. \end{aligned}$$

Lemma 2.13

With assumptions as above, we have an equality

$$\begin{aligned} \chi _{{\mathcal {S}}, T} = 1 - \#T + \chi (F(S){/}F^+, {{\mathrm{ad}}}\overline{r}) - \sum _{v \in S} \chi (F_v^+, {{\mathrm{ad}}}\overline{r}) - \sum _{v \in S - T} \left( \ell _v^1 - n^2 \right) \end{aligned}$$

Proof

Use the above long exact sequence and the exact sequences ($v \in S$):

$$\begin{aligned} 0 \rightarrow H^0(F_v^+, {{\mathrm{ad}}}\overline{r}) \rightarrow H^0(F_v^+, {\mathfrak {g}}_n) \rightarrow H^0(F_v^+, k) \rightarrow H^1(F_v^+, {{\mathrm{ad}}}\overline{r}) \rightarrow H^1(F_v^+, {{\mathrm{ad}}}\overline{r})^\eta \rightarrow 0. \end{aligned}$$

We also note that for $i \ge 3$, the map

$$\begin{aligned} H^i(F(S){/}F^+, {{\mathrm{ad}}}\overline{r}) \rightarrow \prod _{v \in S_\infty } H^i(F_v^+, {{\mathrm{ad}}}\overline{r}) \end{aligned}$$

is bijective (and so $H^i_{{\mathcal {S}}, T} = 0$ if $i \ge 4$). $\square$

Corollary 2.14

Suppose that $S = T \sqcup Q \sqcup S_\infty$ (a disjoint union) and that $S_p \subset T$. Then we have

$$\begin{aligned} \chi _{{\mathcal {S}}, T} = 1 - \# T - \sum _{v \in Q \sqcup S_\infty } (\ell _v^1 - n^2). \end{aligned}$$

Proof

We evaluate each term of the formula of Lemma 2.13 in turn. We have

$$\begin{aligned} \chi (F(S){/}F^+, {{\mathrm{ad}}}\overline{r}) = \sum _{v \in S_\infty } \left( h^0(F_v^+, {{\mathrm{ad}}}\overline{r}) - n^2 \right) \end{aligned}$$

(by [23, Theorem 5.1]), and

$$\begin{aligned} \sum _{v \in S_\infty } \chi (F_v^+, {{\mathrm{ad}}}\overline{r}) = \sum _{v \in S_\infty } h^0(F_v^+, {{\mathrm{ad}}}\overline{r}) \end{aligned}$$

(as $h^1(F_v^+, {{\mathrm{ad}}}\overline{r}) = h^2(F_v^+, {{\mathrm{ad}}}\overline{r})$), and

$$\begin{aligned} \sum _{v \in T \sqcup Q} \chi (F_v^+, {{\mathrm{ad}}}\overline{r}) = -n^2[F^+{:}\,{\mathbb {Q}}] \end{aligned}$$

(by [23, Theorem 2.8]). Summing these up now gives

as desired. $\square$

If $v \in S$, then we define

$$\begin{aligned} \mu _v = \ker \left( H^1(F_v^+, {{\mathrm{ad}}}\overline{r}) \rightarrow H^1(F_v^+, {\mathfrak {g}}_n \overline{r})\right) . \end{aligned}$$

If v is split in F, then $\mu _v = 0$, but in general it can be non-trivial. We always have $\mu _v \subset {\mathcal {L}}_v$. If $T \subset S$, then we define a ‘dual Selmer group’

$$\begin{aligned} H^1_{{\mathcal {S}}^\perp , T} = \left\{ x \in H^1(F(S){/}F^+, {{\mathrm{ad}}}\overline{r}(1)) \mid \text { }\forall v \in T, \langle x, \mu _v \rangle = 0; \forall v \in S - T, \langle x, {\mathcal {L}}_v \rangle = 0 \right\} . \end{aligned}$$

We define $h^1_{{\mathcal {S}}^\perp , T} = \dim _k H^1_{{\mathcal {S}}^\perp , T}$.

Lemma 2.15

Let ${\mathcal {S}}= (F, \overline{r}, {\mathcal {O}}, \chi , S, \{ {\mathcal {D}}_v \}_{v \in S})$ be a global deformation problem, and let $T \subset S$. Then $h^2_{{\mathcal {S}}, T} = h^1_{{\mathcal {S}}^\perp , T}$ and $h^3_{{\mathcal {S}}, T} = h^0(F(S){/}F^+, {{\mathrm{ad}}}\overline{r}(1))$.

Proof

We use the Poitou–Tate exact sequence. More precisely, we have two exact sequences of k-vector spaces:

(2)

and

(3)

The first of these is part of the long exact sequence (1), while the second arises from the Poitou–Tate exact sequence (see [23, Theorem 4.10]). The lemma now follows immediately on comparing (2) and (3). $\square$

2.3 Local deformation problems

We continue with the notation of Sect. 2.1, and define some local deformation problems.

2.3.1 Deformations at infinity

Suppose that $v \in S_\infty$ and $\chi (c_v) = -1$ (we will always assume this in applications below). In this section, we study the unrestricted deformation functor ${{\mathrm{Lift}}}_v^\square$ and its tangent space $\ell ^1_v = {{\mathrm{Lift}}}_v^\square (k[\epsilon ])$.

Lemma 2.16

Suppose that $p = 2$.

• (i) Suppose that n is odd. Then $\overline{r}(c_v)$ is ${{\mathrm{GL}}}_n(k)$-conjugate to $(1_n, 1)\jmath$.

• (ii) Suppose that n is even. Then $\overline{r}(c_v)$ is ${{\mathrm{GL}}}_n(k)$-conjugate either to $(1_n, 1)\jmath$ or $(\Psi _n, 1) \jmath$, where $\Psi _n \in {{\mathrm{GL}}}_n(k)$ is the matrix with 1’s on the antidiagonal and 0’s everywhere else.

Proof

Let $\overline{r}(c_v) = (\overline{A}, 1)\jmath$. We calculate

$$\begin{aligned} \overline{r}(c_v)^2 = (\overline{A} {}^t \overline{A}^{-1}, 1) = 1, \end{aligned}$$

hence $\overline{A} = {}^t \overline{A}$. Conjugation by $g \in {{\mathrm{GL}}}_n(k)$ replaces $\overline{A}$ by $g \overline{A}{}^t g$, so the problem comes down to the classification of ${{\mathrm{GL}}}_n(k)$-conjugacy classes of symmetric matrices.

Rephrasing the problem slightly, we must show that for any $m \ge 1$, any k-vector space V of dimension m with non-degenerate symmetric bilinear pairing is isomorphic to one of $A_m$ or $B_m$, where $A_m = k^m$ with the pairing of Gram matrix $1_n$, and $B_m = k^m$ with the pairing of Gram matrix $\Psi _m$. This follows by induction, together with the easily checked observations $A_m \cong A_1^m,\,B_m \cong B_2^{m{/}2}$ or $B_2^{(m-1){/}2} \oplus A_1$ (according to whether m is even or odd, respectively) and $B_2 \oplus A_1 \cong A_3$. $\square$

Lemma 2.17

• (i) We have $\ell ^1_v = n(n+1){/}2$.

• (ii) Suppose that $p \ne 2$ or that $p = 2$ and $\overline{r}(c_v)$ is ${{\mathrm{GL}}}_n(k)$-conjugate to $(1_n, 1)\jmath$. Then $\dim _k \mu _v = 1$ and the natural map $H^1(F_v^+, k) \rightarrow H^1(F_v^+, {{\mathrm{ad}}}\overline{r})$ is injective.

• (iii) Suppose that $p = 2,\,n$ is even, and $\overline{r}(c_v)$ is ${{\mathrm{GL}}}_n(k)$-conjugate to $(\Psi _n, 1)\jmath$. Then $\dim _k \mu _v = 0$ and the natural map $H^1(F_v^+, k) \rightarrow H^1(F_v^+, {{\mathrm{ad}}}\overline{r})$ is 0.

Proof

Let $r \in {{\mathrm{Lift}}}_v^\square (k[\epsilon ])$. Then we can write $r(c_v) = ( ( 1 + \epsilon X) \overline{A}, 1)\jmath$, and $X \in {{\mathrm{ad}}}\overline{r}$ is arbitrary, subject to the condition

$$\begin{aligned} r(c_v)^2= & {} \left( ( 1 + \epsilon X) \overline{A}, 1\right) \jmath \left( ( 1 + \epsilon X)\overline{A}, 1\right) \jmath \\= & {} \left( ( 1 + \epsilon X) \overline{A}, 1\right) \left( {}^t\overline{A}^{-1} ( 1 - \epsilon {}^t X), 1\right) = (1 + \epsilon ( X - {}^t X), 1) = 1. \end{aligned}$$

In other words, X must be symmetric. This proves the first part of the lemma. The second part is clear if $p \ne 2$. If $p = 2$, we have

$$\begin{aligned} \mu _v = \ker (H^1(F_v^+, {{\mathrm{ad}}}\overline{r}) \rightarrow H^1\left( F_v^+, {\mathfrak {g}}_n \overline{r})\right) = \text {im}\left( H^0(F_v^+, k) \rightarrow H^1(F_v^+, {{\mathrm{ad}}}\overline{r})\right) , \end{aligned}$$

where the boundary map is the one attached to the short exact sequence

A calculation with cocycles reveals that we can identify $\mu _v$ with the image of the map $H^1(F_v^+, k) \rightarrow H^1(F_v^+, {{\mathrm{ad}}}\overline{r})$ which is attached to the short exact sequence

Let us assume that $\overline{r}(c_v)$ is equal either to $(1_n, 1) \jmath$ or $(\Psi _n, 1) \jmath$. Let ${\mathfrak {t}}\subset {{\mathrm{ad}}}\overline{r}$ denote the diagonal Cartan subalgebra, and consider the exact sequence of $k[G_{F_v^+}]$-modules

Then ${\mathfrak {t}}\subset {{\mathrm{ad}}}\overline{r}$ admits a $G_{F_v^+}$-stable complement (consisting of matrices with 0’s on the diagonal), so we can finish the proof of the lemma by analysing what happens inside ${\mathfrak {t}}$.

If $\overline{r}(c_v) = (1_n, 1) \jmath$ (so we are in the second case of the lemma), then $G_{F_v^+}$ acts trivially on each term in this sequence, which is therefore a split exact sequence of $k[G_{F_v^+}]$-modules. The desired assertions follow immediately from this. If $\overline{r}(c_v) = (\Psi _n, 1)\jmath$ (so we are in the third case of the lemma), then $c_v$ acts on ${\mathfrak {t}}\cong k^{2n}$ by reversing the order of coordinates. The usual calculation of cohomology of cyclic groups shows that $H^1(F_v^+, {\mathfrak {t}}) = 0$, which implies the desired statement in this case also. $\square$

2.3.2 Taylor–Wiles deformations

Let v be a finite place of $F^+$ which splits in $F(\zeta _p)$ and at which $\overline{r}$ is unramified, and suppose that $\overline{r}({{\mathrm{Frob}}}_v)$ is semi-simple. Let $\alpha _v \in k$ be an eigenvalue of $\overline{r}({{\mathrm{Frob}}}_v)$ of multiplicity $n_1$, say. We can decompose

$$\begin{aligned} \overline{\rho }|_{G_{F_v^+}} = \overline{A}_v \oplus \overline{B}_v, \end{aligned}$$

(4)

where $A_v({{\mathrm{Frob}}}_v) = \alpha _v \cdot 1_{n_1}$. We define a subfunctor ${\mathcal {D}}_v^\text {TW} \subset {{\mathrm{Lift}}}_v^\square$ as follows: if $R \in {\mathrm {CNL}}_{\mathcal {O}}$ and $r \in {{\mathrm{Lift}}}_v^\square (R)$, then we say that $r \in {\mathcal {D}}_v^\text {TW}(R)$ if there is a decomposition

$$\begin{aligned} \rho = A_v \oplus B_v \end{aligned}$$

(5)

lifting the decomposition (4), and such that $B_v$ is unramified and $A_v|I_{F_v^+} = \psi _v \cdot 1_{n_1}$ for some character $\psi _v{:}\,I_{F_v^+} \rightarrow R^\times$. The deformation problem ${\mathcal {D}}_v^\text {TW}$ depends on the choice of $\alpha _v$, although we do not include it in the notation. It is known (see [32, Lemma 4.2]) that ${\mathcal {D}}_v^\text {TW}$ is a local deformation problem.

Let $\Delta _v = k(v)^\times (p)$ denote the p-part of the finite abelian group $k(v)^\times$. We observe that if $\rho \in {\mathcal {D}}_v^\text {TW}(R)$, then there is a canonical homomorphism $\Delta _v \rightarrow R^\times$, given by $\psi _v \circ {{\mathrm{Art}}}_{F_v^+}$.

2.3.3 Potentially crystalline deformations

Now suppose that $v \in S_p$; then v splits in F, by assumption. We now recall, following [5, § 1.4], some local deformation problems whose existence and basic properties have been established by Kisin [17]. Let ${\mathbb {Z}}^n_+ \subset {\mathbb {Z}}^n$ denote the set of tuples $(\lambda _1, \ldots , \lambda _n)$ with $\lambda _1 \ge \cdots \ge \lambda _n$, and let ${\varvec{\lambda }}_v \in ({\mathbb {Z}}_n^+)^{{{\mathrm{Hom}}}_{{\mathbb {Q}}_p}(F_v^+, \overline{{\mathbb {Q}}}_p)}$. Let K be a finite extension of $F_v^+$ inside $\overline{F}_v^+$.

According to [5, § 1.4], there is a reduced, p-torsion free quotient $R_v^{{\varvec{\lambda }}_v, K-\text {cr}}$ of $R_v^\square$ with the following properties:

• The quotient $R_v^\square \rightarrow R_v^{{\varvec{\lambda }}_v, K-\text {cr}}$ defines a local deformation problem. We write ${\mathcal {D}}_v^{{\varvec{\lambda }}_v, K-\text {cr}}$ for the corresponding set-valued functor in ${\mathrm {CNL}}_{\mathcal {O}}$. In the case $K = F_v^+$, we omit the superscript from the notation and simply write ${\mathcal {D}}_v^{{\varvec{\lambda }}_v}$.

• Let $\rho ^\square {:}\,G_{F^+_v} \rightarrow {{\mathrm{GL}}}_n(R_v^\square )$ denote the universal lifting, and fix a homomorphism $f{:}\,R_v^\square \rightarrow \overline{{\mathbb {Q}}}_p$. Then f factors through the quotient $R_v^\square \rightarrow R_v^{{\varvec{\lambda }}_v, K-\text {cr}}$ if and only if $f \circ \rho ^\square |_{G_K}$ is crystalline, of Hodge–Tate type ${\varvec{\lambda }}_v$.

• The ring $R_v^{{\varvec{\lambda }}_v, K-\text {cr}}[1{/}p]$ is formally smooth over E, of dimension $1 + n^2 + [F_v^+{:}\,{\mathbb {Q}}_p]n(n-1){/}2$.

The ring $R_v^{{\varvec{\lambda }}_v, K-\text {cr}} = R_v^{{\varvec{\lambda }}_v, K-\text {cr}} \otimes _{\mathcal {O}}\overline{{\mathbb {Q}}}_p$ is independent of the choice of coefficient field E (see [5, Lemma 1.2.1]). This fact will be used in Sect. 3.2.

2.3.4 A useful local deformation problem

Now suppose that $v \in S - (S_p \cup S_\infty )$ splits in F. We write $R_v^{\text {fl}}$ for the maximal reduced, p-torsion free quotient of $R_v^\square [1{/}p]$, and ${\mathcal {D}}_v^{\text {fl}}$ for the corresponding set-valued functor on ${\mathrm {CNL}}_{\mathcal {O}}$. It is easy to see that ${\mathcal {D}}_v^{\text {fl}}$ is a local deformation problem.

The ring $R_v^{\text {fl}}$ has the following useful property (see [5, Lemma 1.3.2]). Let $\rho ^\square {:}\,G_{F_v^+} \rightarrow {{\mathrm{GL}}}_n(R_v^\square )$ be the universal lifting, and let $f{:}\,R_v^\square \rightarrow \overline{{\mathbb {Q}}}_p$ be a homomorphism (that therefore factors through the quotient $R_v^{\text {fl}}$). Then there is an irreducible admissible representation $\pi$ of ${{\mathrm{GL}}}_n(F_v^+)$ over $\overline{{\mathbb {Q}}}_p$ such that ${{\mathrm{rec}}}_{F_v^+}^T(\pi ) \cong (f \circ \rho ^\square )^{\text {F-ss}}$. Suppose that $\pi$ is generic. Then the ring $R_v^{\text {fl}}[1{/}p]$ is formally smooth over E at the closed point corresponding to f.

The ring $R_v^{\text {fl}} = R_v^{\text {fl}} \otimes _{\mathcal {O}}\overline{{\mathbb {Q}}}_p$ is independent of the choice of coefficient field E (see [5, Lemma 1.2.1]). This fact will be used in Sect.  3.2.

2.4 Taylor–Wiles systems

Consider a global deformation problem

$$\begin{aligned} {\mathcal {S}}= \left( F, \overline{r}, {\mathcal {O}}, \chi , S, \{ {\mathcal {D}}_v \}_{v \in S}\right) . \end{aligned}$$

Let $T = S - S_\infty$, and $\overline{\rho } = \overline{r}|_{G_{F, S}}$.

Definition 2.18

Let $N \ge 1$ be an integer. A Taylor–Wiles datum of level N is a pair $(Q, ( \alpha _v )_{v \in Q})$ satisfying the following conditions:

• (i) Q is a finite set of finite places of $F^+$.

• (ii) For each $v \in Q,\,v \not \in S$ and v splits in $F(\zeta _{p^N})$.

• (iii) For each $v \in Q,\,\overline{\rho }({{\mathrm{Frob}}}_v)$ is semi-simple and $\alpha _v \in k$ is one of its eigenvalues.

If $(Q, ( \alpha _v )_{v \in Q})$ is a Taylor–Wiles datum, then we define the augmented global deformation problem

$$\begin{aligned} {\mathcal {S}}_Q = \left( F, \overline{r}, {\mathcal {O}}, \chi , S \cup Q, \{ {\mathcal {D}}_v \}_{v \in S} \cup \{ {\mathcal {D}}_v^\text {TW} \}_{v \in Q}\right) , \end{aligned}$$

where for each $v \in Q$ the local deformation problem ${\mathcal {D}}_v^\text {TW}$ is defined with respect to the eigenvalue $\alpha _v \in k$, as in Sect. 2.3.2. Let $\Delta _Q = \prod _{v \in Q} k(v)^\times (p)$. Then there is a canonical homomorphism ${\mathcal {O}}[\Delta _Q] \rightarrow R_{{\mathcal {S}}_Q}$, and a canonical identification $R_{{\mathcal {S}}_Q} \otimes _{{\mathcal {O}}[\Delta _Q]} {\mathcal {O}}= R_{\mathcal {S}}$.

Lemma 2.19

Let $(Q, ( \alpha _v )_{v \in Q})$ be a Taylor–Wiles datum. Then there is an exact sequence

the last arrow being given by $[\psi ] \mapsto ({{\mathrm{tr}}}e_{{{\mathrm{Frob}}}_v, \alpha _v} \psi ({{\mathrm{Frob}}}_v))_{v \in Q}$. (Here we write $e_{{{\mathrm{Frob}}}_v, \alpha _v} \in {{\mathrm{ad}}}\overline{r} = M_n(k)$ for the unique idempotent in $k[\overline{\rho }({{\mathrm{Frob}}}_v)]$ with image equal to the $\alpha _v$-eigenspace of $\overline{\rho }({{\mathrm{Frob}}}_v))$.

Proof

If $v \in Q$, let ${\mathcal {L}}_v \subset H^1(F_v^+, {{\mathrm{ad}}}\overline{r})$ be the subspace corresponding to ${\mathcal {D}}_v^\text {TW}(k[\epsilon ])$, and let ${\mathcal {L}}_v^\text {ur}$ be the subspace corresponding to the functor of unramified deformations. Then ${\mathcal {L}}_v^\text {ur} \subset {\mathcal {L}}_v$. By definition, we have

$$\begin{aligned} H^1_{{\mathcal {S}}^\perp , T}= & {} \ker \Big [ H^1(F(S \cup Q){/}F^+, {{\mathrm{ad}}}\overline{r}(1) ) \rightarrow \\&\quad \prod _{v \in T} H^1(F_v^+, {{\mathrm{ad}}}\overline{r}(1)){/}\mu _v^\perp \times \prod _{v \in S - T} H^1(F_v^+, {{\mathrm{ad}}}\overline{r}(1) ){/}{\mathcal {L}}_v^\perp \\&\qquad \times \prod _{v \in Q} H^1(F_v^+, {{\mathrm{ad}}}\overline{r}(1)){/}({\mathcal {L}}_v^\text {ur})^\perp \Big ], \end{aligned}$$

and

$$\begin{aligned} H^1_{{\mathcal {S}}_Q^\perp , T} = \ker H^1_{{\mathcal {S}}^\perp , T} \rightarrow \prod _{v \in Q} ({\mathcal {L}}_v^\text {ur})^\perp {/}{\mathcal {L}}_v^\perp . \end{aligned}$$

To show the lemma, it is therefore enough to show that for each $v \in Q$, there is an isomorphism $({\mathcal {L}}_v^\text {ur})^\perp {/}{\mathcal {L}}_v^\perp \cong k$, given at the level of cocycles by the formula $[\psi ] \mapsto {{\mathrm{tr}}}e_{{{\mathrm{Frob}}}_v, \alpha _v} \psi ({{\mathrm{Frob}}}_v)$. Writing $\overline{r}|_{G_{F_v^+}} = \overline{A}_v \oplus \overline{B}_v$ as in the definition of ${\mathcal {D}}_v^\text {TW}$, we calculate ${{\mathrm{ad}}}\overline{r} \cong {{\mathrm{End}}}(\overline{A}_v) \oplus {{\mathrm{Hom}}}(\overline{A}_v, \overline{B}_v) \oplus {{\mathrm{Hom}}}(\overline{B}_v, \overline{A}_v) \oplus {{\mathrm{End}}}(\overline{B}_v)$, hence

$$\begin{aligned} H^1(F_v^+, {{\mathrm{ad}}}\overline{r}) \cong H^1(F_v^+, {{\mathrm{ad}}}\overline{A}_v) \oplus H^1(F_v^+, {{\mathrm{ad}}}\overline{B}_v) \end{aligned}$$

(as the other two summands have trivial cohomology). The representation $\overline{A}_v$ is assumed to be scalar, so ${{\mathrm{ad}}}\overline{A}_v$ is a trivial Galois module, and we have a canonical isomorphism $H^1(F_v^+, {{\mathrm{ad}}}\overline{A}_v) \cong H^1(F_v^+, k) \otimes _k {{\mathrm{ad}}}\overline{A}_v$. The group $H^1(F_v^+, k)$ is a 2-dimensional k-vector space, endowed with a perfect duality; the subspace of unramified classes is 1-dimensional, and equal to its own orthogonal complement. Writing $Z \subset {{\mathrm{ad}}}\overline{A}_v$ for the subspace of scalar endomorphisms, we can therefore identify

$$\begin{aligned} {\mathcal {L}}_v^\perp= & {} \left\{ (x, y) \in {\mathcal {L}}_v^\text {ur} \subset H^1(F_v^+, {{\mathrm{ad}}}\overline{A}_v) \oplus H^1(F_v^+, {{\mathrm{ad}}}\overline{B}_v)\right. \\\cong & {} \left. H^1(F_v^+, {{\mathrm{ad}}}\overline{r}) \mid \langle x, H^1(F_v^+, Z) \rangle = 0 \right\} \\= & {} H^1_\text {ur}(F_v^+, k) \otimes _k {{\mathrm{ad}}}^0 \overline{A}_v \oplus H^1_\text {ur}(F_v^+, \overline{B}_v). \end{aligned}$$

This expression is equivalent to the desired result. $\square$

The following definition of adequate subgroups is taken from [13].

Definition 2.20

Let K be a field. We say that a subgroup $H \subset {{\mathrm{GL}}}_n(K)$ is adequate if it satisfies the following conditions:

• (i) We have $H^1(H, K) = 0$ and $H^1(H, {{\mathrm{ad}}}_0) = 0$.

• (ii) For each simple K[H]-submodule $W \subset {{\mathrm{ad}}}$, there exists a semi-simple element $\sigma \in H$ with an eigenvalue $\alpha \in K$ such that ${{\mathrm{tr}}}e_{\sigma , \alpha } W \ne 0$.

The second condition implies that $M_n(K)$ is spanned as a K-vector space by the semi-simple elements $\sigma \in H \subset {{\mathrm{ad}}}$. In particular, an adequate subgroup acts absolutely irreducibly in its tautological representation on $K^n$.

Proposition 2.21

Let ${\mathcal {S}}= (F, \overline{r}, {\mathcal {O}}, \chi , S, \{ {\mathcal {D}}_v \}_{v \in S})$ be a global deformation problem, and let $T = S - S_\infty$. We make the following assumptions:

• (i) For each $v \in S_\infty ,\,\mu (c_v) = -1$ and ${\mathcal {D}}_v = {{\mathrm{Lift}}}_v^\square$.

• (ii) If $p \ne 2$, then $F = F^+(\zeta _p)$. If $p = 2$, then $F = F^+(\sqrt{-1})$.

• (iii) If $p = 2$ and n is even, then there exists $v \in S_\infty$ such that $\overline{r}(c_v)$ is ${{\mathrm{GL}}}_n(k)$-conjugate to $(1_n, 1)\jmath$ (cf. Sect. 2.3.1).

• (iv) The group $\overline{\rho }(G_F) \subset {{\mathrm{GL}}}_n(k)$ is adequate.

Let $q = h^1_{{\mathcal {S}}^\perp , T} - 1$ and $g = q + |T| - 1 - [F^+{:}\,{\mathbb {Q}}]n(n-1){/}2$. Then for each $N \ge 1$, we can find infinitely many Taylor–Wiles data $(Q, ( \alpha _v )_{v \in Q})$ of level N such that $\# Q = q$ and the map $R_{{\mathcal {S}}, T}^\text {loc} \rightarrow R_{{\mathcal {S}}_Q}^T$ can be extended to a surjection $R_{{\mathcal {S}}, T}^\text {loc}\llbracket X_1, \ldots , X_{g} \rrbracket \twoheadrightarrow R_{{\mathcal {S}}_Q}^T$.

Proof

We first note that $R_{{\mathcal {S}}, T}^\text {loc} = R_{{\mathcal {S}}_Q, T}^\text {loc}$ by definition, and that $R_{{\mathcal {S}}_Q}^T$ can be topologically generated as an $R_{{\mathcal {S}}_Q, T}^\text {loc}$-algebra by $h^1_{{\mathcal {S}}_Q, T}$ elements. To prove the proposition, it is therefore enough to show that for each $N \ge 1$, we can find infinitely many Taylor–Wiles data $(Q, ( \alpha _v )_{v \in Q})$ of level N such that $|Q| = q$ and $h^1_{{\mathcal {S}}_Q, T} = g$.

On the other hand, we have by Corollary 2.14 and the local calculations of Sect. 2.3 an equality

$$\begin{aligned} \chi _{{\mathcal {S}}_Q, T} = 1 - |T| - |Q| + [F^+{:}\,{\mathbb {Q}}] n(n-1){/}2. \end{aligned}$$

Combining this with Lemma 2.15, we obtain an equality

$$\begin{aligned} h^1_{{\mathcal {S}}_Q, T}= & {} h^2_{{\mathcal {S}}_Q, T} - h^3_{{\mathcal {S}}_Q, T} - 1 + |T| + |Q| - [F^+{:}\,{\mathbb {Q}}] n(n-1){/}2\\= & {} h^1_{{\mathcal {S}}_Q^\perp , T} - 2 + |T| + |Q| - [F^+{:}\,{\mathbb {Q}}] n(n-1){/}2. \end{aligned}$$

It is therefore enough to show that for each $N \ge 1$, we can find infinitely many Taylor–Wiles data $(Q, ( \alpha _v )_{v \in Q})$ of level N such that $h^1_{{\mathcal {S}}_Q^\perp , T} = 1 = h^1_{{\mathcal {S}}^\perp , T} - |Q|$. We will do this by killing cohomology classes in $H^1_{{\mathcal {S}}^\perp , T}$ in the usual manner, using Lemma 2.19.

Fix now a choice of $N \ge 2$, large enough so that $F_N = F(\zeta _{p^N})$ strictly contains F. Let $[\psi ] \in H^1_{{\mathcal {S}}^\perp , T} \subset H^1(F(S){/}F^+, {{\mathrm{ad}}}\overline{r}(1))$ be a cohomology class with non-zero image in $H^1(F(S){/}F_N, {{\mathrm{ad}}}\overline{r}(1))$. We claim that we can find infinitely many Taylor–Wiles data $(\{ w \}, \alpha _w)$ of level N such that $[\psi ] \not \in H^1_{{\mathcal {S}}_{\{ w \}}^\perp , T}$.

We first show that this claim implies the proposition. Indeed, let s denote the dimension of the image of $H^1_{{\mathcal {S}}^\perp , T}$ in $H^1(F(S){/}F_N, {{\mathrm{ad}}}\overline{r}(1))$. By applying the claim repeatedly we can find infinitely many Taylor–Wiles data $(Q, \{ \alpha _v \}_{v \in Q})$ of level N such that $|Q| = s,\,h^1_{{\mathcal {S}}_Q^\perp , T} = h^1_{{\mathcal {S}}^\perp , T} - s$ and the map $H^1_{{\mathcal {S}}^\perp _Q, T} \rightarrow H^1(F(S){/}F_N, {{\mathrm{ad}}}\overline{r}(1))$ is trivial.

It follows that $H^1_{{\mathcal {S}}^\perp _Q, T}$ is contained inside

$$\begin{aligned} \ker \left[ H^1(F(S){/}F^+, {{\mathrm{ad}}}\overline{r}(1)) \rightarrow H^1(F(S){/}F_N, {{\mathrm{ad}}}\overline{r}(1)) \right] = H^1\left( F_N{/}F^+, {{\mathrm{ad}}}\overline{r}(1)^{G_{F_N}}\right) . \end{aligned}$$

Our assumption that $\overline{\rho }(G_F)$ is adequate implies that $\overline{\rho }(G_{F_N}) = \overline{\rho }(G_F)$ and ${{\mathrm{ad}}}\overline{r}(1)^{G_{F_N}} = k$ (i.e. the subspace of scalar matrices). At this point, we split into cases according to the parity of p. If p is odd, then it is easy to see that $H^1_{{\mathcal {S}}^\perp _Q, T} = H^1(F_N{/}F^+, k)$ is 1-dimensional. It follows that $s = q = |Q|$, and the proof of the proposition is completed in this case. If $p = 2$, then $H^1(F_N{/}F^+, k)$ is 2-dimensional. However, it follows from our hypotheses and the second part of Lemma 2.17 that there exists a place $v \in S_\infty$ such that the map $H^1(F_N{/}F^+, k) \rightarrow H^1(F_v^+, k) \rightarrow H^1(F_v^+, {{\mathrm{ad}}}\overline{r}(1))$ is injective, and hence that $H^1_{{\mathcal {S}}^\perp _Q, T}$ is again 1-dimensional (we observe that classes in $H^1_{{\mathcal {S}}_Q^\perp , T}$ are locally trivial at $v \in S_\infty$, by definition, because $v \not \in T$). We again see that $s = q = |Q|$, and the proof of the proposition is completed in this case also.

We now return to the proof of the claim. Let $[\psi ] \in H^1_{{\mathcal {S}}^\perp , T} \subset H^1(F(S){/}F^+, {{\mathrm{ad}}}\overline{r}(1))$ be a cohomology class with non-zero image in $H^1(F(S){/}F_N, {{\mathrm{ad}}}\overline{r}(1))$. By Lemma 2.19, it is enough to find a place w of $F^+$ and an element $\alpha _w \in k$ satisfying the following conditions:

• w splits in $F_N$ and $\overline{\rho }({{\mathrm{Frob}}}_w)$ is semi-simple.

• $\alpha _w \in k$ is an eigenvalue of $\overline{\rho }({{\mathrm{Frob}}}_w)$ such that ${{\mathrm{tr}}}e_{{{\mathrm{Frob}}}_w, \alpha _w} \psi ({{\mathrm{Frob}}}_w) \ne 0$.

By the Chebotarev density theorem, it is even enough to find elements $\sigma \in G_{F_N}$ and $\alpha \in k$ such that $\overline{\rho }(\sigma )$ is semi-simple, and $\alpha$ is an eigenvalue of $\overline{\rho }(\sigma )$ such that ${{\mathrm{tr}}}e_{\sigma , \alpha } \psi (\sigma ) \ne 0$.

Let K / F be the extension cut out by ${{\mathrm{ad}}}\overline{\rho }$, and let $K_N = K \cdot F_N$. Let $f = \psi |_{K_N}$, an element of $H^1(F(S){/}K_N, {{\mathrm{ad}}}\overline{r}(1))^{G_F} \subset H^1(F(S){/}K_N, {{\mathrm{ad}}}\overline{\rho })^{G_{F_N}}$. The image of $[\psi ]$ in $H^1(F(S){/}F_N, {{\mathrm{ad}}}\overline{r})$ is non-zero. On the other hand, the group $H^1(K_N{/}F_N, {{\mathrm{ad}}}\overline{r}) \cong H^1(K{/}F, {{\mathrm{ad}}}\overline{\rho }) = 0$, as follows from the assumption that $\overline{\rho }$ has adequate image. It follows that f is non-zero, as a homomorphism $f{:}\,{{\mathrm{Gal}}}(F(S){/}K_N) \rightarrow {{\mathrm{ad}}}\overline{r}$.

Let $V \subset {{\mathrm{ad}}}\overline{\rho }$ denote the k-span of the image of f, a $k[G_{F_N}]$-module. We can find a simple $k[G_{F_N}]$-submodule $W \subset V$, an element $\sigma _0 \in G_{F_N}$ such that $\overline{\rho }(\sigma _0)$ is semi-simple, and an eigenvalue $\alpha _0 \in k$ of $\overline{\rho }(\sigma _0)$ such that ${{\mathrm{tr}}}e_{\sigma _0, \alpha _0} W \ne 0$. If ${{\mathrm{tr}}}e_{\sigma _0, \alpha _0} \psi (\sigma _0) \ne 0$, then we are done on taking $\sigma = \sigma _0$ and $\alpha = \alpha _0$.

Let us suppose instead that ${{\mathrm{tr}}}e_{\sigma _0, \alpha _0} \psi (\sigma _0) = 0$, and choose an element $\tau \in K_N$ such that ${{\mathrm{tr}}}e_{\sigma _0, \alpha _0} f(\tau ) \ne 0$. We claim that we are now done on taking $\sigma = \tau \sigma _0$. Indeed, there is an eigenvalue $\alpha$ of $\overline{\rho }(\sigma )$ such that $e_{\sigma _0, \alpha _0} = e_{\sigma , \alpha }$, and we calculate using the cocycle relation

$$\begin{aligned} {{\mathrm{tr}}}e_{\sigma , \alpha } \psi (\sigma ) = {{\mathrm{tr}}}e_{\sigma _0, \alpha _0} (\psi (\tau ) + \psi (\sigma _0)) = {{\mathrm{tr}}}e_{\sigma _0, \alpha _0} f(\tau ) \ne 0. \end{aligned}$$

This completes the proof. $\square$

3 Galois representations and automorphic representations of ${{\mathrm{GL}}}_n({\mathbb {A}}_F)$

In this section, we recall the class of automorphic representations with which we work and make some preliminary observations about their attached Galois representations.

Definition 3.1

Let $n \ge 1$ be an integer.

• (i) Let F be an imaginary CM number field. A RACSDC automorphic representation is an automorphic representation $\pi$ of ${{\mathrm{GL}}}_n({\mathbb {A}}_F)$ satisfying the following conditions:

  • (a) $\pi$ is regular algebraic.

  • (b) $\pi$ is conjugate self-dual, i.e. $\pi ^c \cong \pi ^\vee ,\,c \in {{\mathrm{Gal}}}(F{/}F^+)$ the non-trivial element.

  • (c) $\pi$ is cuspidal.

• (ii) Let F be a totally real number field. A RAESDC automorphic representation is a pair $(\pi , \chi )$ consisting of an automorphic representation $\pi$ of ${{\mathrm{GL}}}_n({\mathbb {A}}_F)$ and a continuous character $\chi {:}\,{\mathbb {A}}_F^\times \rightarrow {\mathbb {C}}^\times$, satisfying the following conditions:

  • (a) $\pi$ is regular algebraic.

  • (b) $\pi$ is essentially self-dual, i.e. $\pi \cong \pi ^\vee \otimes (\chi \circ \det )$.

  • (c) $\pi$ is cuspidal.

It is usual to require in the definition of a RAESDC automorphic representation that the value $\chi _v(-1)$ is independent of the choice of place $v|\infty$ of F; however, Patrikis [24, Theorem 2.0.1] has recently shown that this follows from the other conditions.

If F is an imaginary CM field or a totally real field, and $\pi$ [resp. $(\pi , \chi )$] is a RACSDC (resp. RAESDC) automorphic representation of ${{\mathrm{GL}}}_n({\mathbb {A}}_F)$, and $\iota {:}\,\overline{{\mathbb {Q}}}_p \cong {\mathbb {C}}$ is an isomorphism, then there exists a continuous semi-simple representation $r_\iota (\pi ){:}\,G_F \rightarrow {{\mathrm{GL}}}_n(\overline{{\mathbb {Q}}}_p)$, which satisfies the following local-global compatibility condition at each finite place v of F:

$$\begin{aligned} \mathrm {WD}(r_\iota (\pi )|_{G_{F_v}})^{\text {F-ss}} \cong {{\mathrm{rec}}}_{F_v}^T( \iota ^{-1} \pi _v ). \end{aligned}$$

In particular, if v | p then $r_\iota (\pi )|_{G_{F_v}}$ is de Rham, and there is a recipe for the Hodge–Tate weights of $\pi$ in terms of $\iota$ and $\pi _\infty$; see [9, § 2.1] for a precise statement and list of references.

3.1 The sign of a conjugate self-dual Galois representation

Let F be an imaginary CM number field with maximal totally real subfield $F^+$, and let $c \in G_{F^+}$ be a fixed choice of complex conjugation. Let k be a field, $\rho {:}\,G_F \rightarrow {{\mathrm{GL}}}_n(k)$ an absolutely irreducible representation, and $\mu {:}\,G_{F^+} \rightarrow k^\times$ a character.

Suppose that there is an isomorphism $\rho ^c \cong \rho ^\vee \otimes \mu$; equivalently, that there exists a pairing $\langle \cdot , \cdot \rangle {:}\,k^n \times k^n \rightarrow k$ such that for all $\delta \in G_F,\,x, y \in k^n$, we have $\langle \rho (\delta ) x, \rho (\delta ^c) y \rangle = \mu (\delta ) \langle x, y \rangle$. The pairing $\langle \cdot , \cdot \rangle$ is then uniquely determined up to scalar, and (after possibly replacing $\mu$ by $\mu \delta _{F{/}F^+}$, as in the proof of Lemma 2.2) we have

$$\begin{aligned} \langle x, y \rangle = - \mu (c) \langle y, x \rangle \end{aligned}$$

(6)

for all $x, y \in k^n$. Following [5, § 2], we say that a pair $(\rho , \mu )$ satisfying the condition (6) is polarized. By Lemma 2.1, this is equivalent to asking that $\rho$ extend to a homomorphism $r{:}\,G_{F^+} \rightarrow {\mathcal {G}}_n(k)$ such that $\nu \circ r = \mu$. This condition is independent of the choice of c. We have the following result, due to Bellaiche–Chenevier [2].

Theorem 3.2

Let $\pi$ be a RACSDC automorphic representation of ${{\mathrm{GL}}}_n({\mathbb {A}}_F),\,\iota {:}\,\overline{{\mathbb {Q}}}_p \cong {\mathbb {C}}$ an isomorphism, and suppose that $r_\iota (\pi )$ is irreducible. Then the pair $(r_\iota (\pi ), \epsilon ^{1-n} \delta _{F{/}F^+}^n)$ is polarized.

Now suppose that k is a perfect field of characteristic 2 and that n is even, and that the pair $(\rho , \mu )$ is polarized. We can attach a discrete invariant to $\rho$ at each infinite place v of $F^+$ as follows. Let $c_v \in G_{F^+}$ be a choice of complex conjugation at this place, and let $r{:}\,G_{F^+} \rightarrow {\mathcal {G}}_n(k)$ be a choice of extension of $\rho$ such that $\nu \circ r = \mu$.

According to Lemma 2.16, there are two distinct possibilities for the ${{\mathrm{GL}}}_n(k)$-conjugacy class of $r(c_v) \in {\mathcal {G}}_n(k)$: it is conjugate either to $(1_n, 1)\jmath$, or to $(\Psi _n, 1)\jmath$. We observe that the matrix $\Psi _n$ admits skew-symmetric lifts in ${{\mathrm{GL}}}_n(W(k))$, while the matrix $1_n$ does not (here W(k) denotes the ring of Witt vectors of k).

Definition 3.3

Let k be a perfect field of characteristic 2, and suppose that n is even. Let $(\rho , \mu )$ be polarized, and let v be an infinite place of $F^+$. We say that the pair $(\rho , \mu )$ is strongly residually odd at v if $r(c_v)$ is ${{\mathrm{GL}}}_n(k)$-conjugate to $(1_n, 1)\jmath$.

To motivate this definition, we have the following simple lemma.

Lemma 3.4

Let $p = 2$, and let n be an even integer. Let E be a finite extension of ${\mathbb {Q}}_p$, and let $r{:}\,G_{F^+} \rightarrow {\mathcal {G}}_n({\mathcal {O}})$ be a homomorphism such that (setting $\rho = r|_{G_F},\,\mu = \nu \circ r)$ $\overline{\rho }$ is absolutely irreducible. Then the pair $(\rho , \mu )$ is polarized. If $v | \infty$ is a place of $F^+$, and $(\overline{\rho }, \overline{\mu })$ is strongly residually odd at v, then $\mu (c_v) = -1$.

Lemma 3.5

Let k be a finite field, and let $\sigma {:}\,G_{F^+} \rightarrow {{\mathrm{GL}}}_2(k)$ be a continuous representation such that $\sigma |_{G_F}$ is absolutely irreducible. Let $\chi = \det \sigma$, and let $\psi {:}\,G_F \rightarrow k^\times$ be a character such that $\psi \psi ^c = \epsilon \chi$, and set $\rho = \sigma |_{G_F} \otimes \psi ^{-1}$. Then:

• (i) The character $\chi$ is totally odd if and only if the pair $(\rho , \epsilon ^{-1})$ is polarized.

• (ii) Suppose that k has characteristic 2, and let v be an infinite place of $F^+$. Then $(\rho , \epsilon ^{-1})$ is strongly residually odd at v if and only if $\sigma (c_v) \in {{\mathrm{GL}}}_2(k)$ is non-trivial.

Proof

The proof is by explicit calculation. Fix a place $v | \infty$ of $F^+$ and a choice $c = c_v \in G_{F^+}$ of complex conjugation at v. Note first that $\rho$ is absolutely irreducible and $\rho ^c \cong \rho ^\vee \otimes \epsilon ^{-1}$. Define

$$\begin{aligned} J = \left( \begin{array}{ll} 0 &{}\quad 1 \\ -1 &{}\quad 0 \end{array}\right) , \end{aligned}$$

so that ${}^t \sigma (\gamma ) J \sigma (\gamma ) = \chi (\gamma ) J$ for all $\gamma \in G_{F^+}$. An easy calculation then shows that the relation

$$\begin{aligned} {}^t \rho (\delta ) A^{-1} \rho (\delta ^c) = \epsilon ^{-1}(\delta ) A^{-1} \end{aligned}$$

holds if we take $A = \sigma (c) J^{-1}$, in which case ${}^t A = - \chi (c) A$. We can therefore define an extension $r{:}\,G_{F^+} \rightarrow {\mathcal {G}}_2(k)$ of $\rho$ by setting $r(c) = (A, \epsilon ^{-1}\chi (c))\jmath$. This shows the first part of the lemma.

For the second part, we observe that the only two possibilities for $\sigma (c_v)$ (up to conjugation in ${{\mathrm{GL}}}_2(k)$) are $\sigma (c_v) = 1$ or

$$\begin{aligned} \sigma (c_v) = \left( \begin{array}{ll} 1 &{}\quad 1 \\ 0 &{}\quad 1 \end{array} \right) . \end{aligned}$$

In the first case, we obtain $A = J^{-1} = \Psi _2$. In the second case, we obtain

$$\begin{aligned} A = \left( \begin{array}{ll} 1 &{}\quad 1 \\ 0 &{}\quad 1 \end{array} \right) \left( \begin{array}{ll} 0 &{}\quad 1 \\ 1 &{}\quad 0 \end{array} \right) = \left( \begin{array}{ll} 1 &{}\quad 1 \\ 1 &{}\quad 0 \end{array} \right) , \end{aligned}$$

and this symmetric matrix is ${{\mathrm{GL}}}_2(k)$-conjugate to $1_2$. This completes the proof. $\square$

3.2 Irreducible components of Galois deformation rings

In this section we recall some important ideas from [5], which formalize the idea of two global Galois representations having the same local properties (compare the statement of Theorem 1.1). Fix a prime p. Let l be a prime, and let K be a finite extension of ${\mathbb {Q}}_l$ inside $\overline{{\mathbb {Q}}}_l$.

Definition 3.6

Fix continuous representations $\rho _1, \rho _2{:}\,G_K \rightarrow {{\mathrm{GL}}}_n(\overline{{\mathbb {Z}}}_p)$, and let $r_1, r_2{:}\,G_K \rightarrow {{\mathrm{GL}}}_n(\overline{{\mathbb {F}}}_p)$ be the corresponding residual representations $r_1 = \rho _1 \text { mod }{\mathfrak {m}}_{\overline{{\mathbb {Z}}}_p},\,r_2 = \rho _2 \text { mod }{\mathfrak {m}}_{\overline{{\mathbb {Z}}}_p}$. Suppose that $r_1 \cong r_2$.

• (i) Suppose that $l \ne p$. We say that $\rho _1$ connects with $\rho _2$, and write $\rho _1 \sim \rho _2$, if $\rho _1, \rho _2$ define points on a common irreducible component of ${{\mathrm{Spec}}}( R_v^{\text {fl}} \otimes \overline{{\mathbb {Q}}}_p )$, where $R_v^{\text {fl}} \otimes \overline{{\mathbb {Q}}}_p$ is the ring defined with respect to $r_1$ in Sect. 2.3.4.

• (ii) Suppose that $l = p$. We say that $\rho _1$ connects with $\rho _2$, and write $\rho _1 \sim \rho _2$, if there exists a finite extension $K'{/}K$ and ${\varvec{\lambda }}_v \in ({\mathbb {Z}}_+^n)^{{{\mathrm{Hom}}}_{{\mathbb {Q}}_p}(K, \overline{{\mathbb {Q}}}_p)}$ such that each of $\rho _1|_{G_{K'}}$ and $\rho _2|_{G_{K'}}$ is crystalline of Hodge–Tate type ${\varvec{\lambda }}_v$, and $\rho _1, \rho _2$ define points on a common irreducible component of ${{\mathrm{Spec}}}( R_v^{{\varvec{\lambda }}_v, K'-\text {cr}} \otimes \overline{{\mathbb {Q}}}_p )$, where $R_v^{{\varvec{\lambda }}_v, K'-\text {cr}} \otimes \overline{{\mathbb {Q}}}_p$ is the ring defined with respect to $r_1$ in Sect. 2.3.3.

(We use the notation $r_1, r_2$ instead of $\overline{\rho }_1, \overline{\rho }_2$ because of our convention that $\overline{\rho }_i$ is the semi-simplified residual representation of $\overline{\rho }$). It follows from [5, Lemma 1.2.2] that these definitions make sense, independently of the choice of isomorphism $r_1 \cong r_2$.

Now let F be a number field, and suppose that $\rho {:}\,G_F \rightarrow {{\mathrm{GL}}}_n(\overline{{\mathbb {Q}}}_p)$ is a continuous representation such that $\overline{\rho }$ is absolutely irreducible. Then after conjugating, we can assume that $\rho$ takes values in ${{\mathrm{GL}}}_n(\overline{{\mathbb {Z}}}_p)$, and $\rho$ is then unique up to ${{\mathrm{GL}}}_n(\overline{{\mathbb {Z}}}_p)$-conjugation.

Suppose $\rho _1, \rho _2{:}\,G_F \rightarrow {{\mathrm{GL}}}_n(\overline{{\mathbb {Q}}}_p)$ are continuous representations such that $\overline{\rho }_1$ and $\overline{\rho }_2$ are absolutely irreducible and $\overline{\rho }_1 \cong \overline{\rho }_2$, and let v be a finite place of F. We will say that $\rho _1|_{G_{F_v}}$ connects to $\rho _2|_{G_{F_v}}$, and write $\rho _1|_{G_{F_v}} \sim \rho _2|_{G_{F_v}}$, if this relation holds as in Definition 3.6 with respect to these choices of integral lattice.

4 Automorphic forms on definite unitary groups

In this section we define spaces of algebraic modular forms on definite unitary groups, and use them to prove an $R = {\mathbb {T}}$ type result. Apart from the Galois-theoretic ingredients of Sect. 2, there is nothing in the current section that will be new to an expert. In order to avoid repetition, we therefore refer to [32, § 6] for detailed definitions and proofs, and merely recall notation and basic properties here. We first fix a prime p and an integer $n \ge 2$. At the beginning of [32, § 6], it is assumed that p is odd, but this plays no role in the parts recalled here. We fix a coefficient field $E \subset \overline{{\mathbb {Q}}}_p$ with ring of integers ${\mathcal {O}}$, maximal ideal $\lambda$, and residue field k.

We begin with an imaginary CM number field F with maximal totally real subfield $F^+$, satisfying the following conditions:

• The extension $F{/}F^+$ is everywhere unramified, and each prime of $F^+$ above p splits in F.

• We have $n[F^+{:}\,{\mathbb {Q}}] \equiv 0 \text { mod }4$.

Let $c \in {{\mathrm{Gal}}}(F{/}F^+)$ denote the non-trivial element. We can then find a unitary group G over $F^+$, split by F, and satisfying the following conditions:

• For each finite place v of $F^+,\,G(F^+_v)$ is quasi-split.

• The group $G(F^+ \otimes _{\mathbb {Q}}{\mathbb {R}})$ is compact.

We can extend G to an affine group scheme over ${\mathcal {O}}_{F^+}$, still denoted G, with the following property: for each place v of $F^+$ split as $v = w w^c$ in F, there is an isomorphism $\iota _w{:}\,G({\mathcal {O}}_{F^+}) \cong {{\mathrm{GL}}}_n({\mathcal {O}}_{F_w})$. We use the following notation:

• $S_p$ is the set of places of $F^+$ above p. For each $v \in S_p$, we choose a place ${\widetilde{v}}$ of F above v and write $\widetilde{S}_p = \{ {\widetilde{v}}\mid v \in S_p \}$. We write $I_p$ for the set of embeddings $F^+ \hookrightarrow \overline{{\mathbb {Q}}}_p$, and $\widetilde{I}_p$ for the set of embeddings $F \hookrightarrow \overline{{\mathbb {Q}}}_p$ inducing a place of $\widetilde{S}_p$.

• Let ${\mathbb {Z}}^+_n = \{ (\lambda _1, \ldots , \lambda _n) \in {\mathbb {Z}}^n \mid \lambda _1 \ge \cdots \ge \lambda _n \}$. For each ${\varvec{\lambda }}= (\lambda _\tau )_\tau \in ({\mathbb {Z}}^+_n)^{\widetilde{I}_p}$, there is a finite free ${\mathcal {O}}$-module $M_{\varvec{\lambda }}$, with a continuous action of the group $\prod _{v \in S_p} G({\mathcal {O}}_{F^+_v}) \cong \prod _{w \in \widetilde{S}_p} {{\mathrm{GL}}}_n({\mathcal {O}}_{F_w})$. It is constructed from the tensor product (over $\tau \in \widetilde{I}_p$) of the algebraic representations of ${{\mathrm{GL}}}_n$ of highest weights $\lambda _\tau$.

• If $U = \prod _v U_v \subset G({\mathbb {A}}_{F^+}^\infty )$ is an open compact subgroup such that $U_v \subset G({\mathcal {O}}_{F^+_v})$ for each $v \in S_p$, and ${\varvec{\lambda }}\in ({\mathbb {Z}}^+_n)$, and A is a ${\mathcal {O}}$-module, then we write $S_{\varvec{\lambda }}(U, A)$ for the set of all functions

  $$\begin{aligned} f{:}\,G(F^+) \backslash G({\mathbb {A}}_{F^+}^\infty ) \rightarrow M_{\varvec{\lambda }}\otimes _{\mathcal {O}}A \end{aligned}$$

  satisfying the following condition: for all $u \in U,\,g \in G({\mathbb {A}}_{F^+}^\infty )$, we have $f ( gu) = u_p^{-1} f(g)$, where $u_p$ denotes projection to the p-component.

• If v is a finite place of $F^+$ and $U_v \subset G(F^+_v)$ is an open compact subgroup, then we write ${\mathcal {H}}(G(F_v^+), U_v)$ for the convolution algebra of $U_v$-biinvariant functions $f{:}\,G(F_v^+) \rightarrow {\mathbb {Z}}$ (with the Haar measure giving $U_v$ total measure 1). If $v = w w^c$ is split in F and $U_v = \iota _w^{-1}({{\mathrm{GL}}}_n({\mathcal {O}}_{F_w}))$, then we write $T_w^i,\,i = 1, \ldots , n$ for the standard unramified Hecke operators given as the characteristic functions of the double cosets

  $$\begin{aligned} U_v \iota _w^{-1}\left( {{\mathrm{diag}}}(\underbrace{\varpi _w, \ldots , \varpi _w}_i, \underbrace{1, \ldots , 1}_{n-i})\right) U_v. \end{aligned}$$

  If $U = \prod _v U_v \subset G({\mathbb {A}}_{F^+}^\infty )$ is an open compact subgroup, and A is an ${\mathcal {O}}$-module, then for each finite place v of $F^+$, the algebra ${\mathcal {H}}(G(F_v^+), U_v)$ acts on $S_{\varvec{\lambda }}(U, A)$ in a natural way.

• If $U = \prod _v U_v \subset G({\mathbb {A}}_{F^+}^\infty )$ is an open compact subgroup and T is a finite set of finite places of $F^+$ such that for each $v \not \in T,\,U_v$ is a hyperspecial maximal compact subgroup and v is prime to p, then we define ${\mathbb {T}}^T_{\varvec{\lambda }}(U, {\mathcal {O}})$ to be the ${\mathcal {O}}$-subalgebra of ${{\mathrm{End}}}_{\mathcal {O}}(S_{\varvec{\lambda }}(U, {\mathcal {O}}))$ generated by the operators $T_w^i,\,i = 1, \ldots , n$, as w ranges through the set of all finite places of F which are split over $F^+$ and prime to T.

• We define

  $$\begin{aligned} {\mathcal {A}}_{\varvec{\lambda }}= \mathop {\varinjlim }\limits _U S_{\varvec{\lambda }}(U, \overline{{\mathbb {Q}}}_p), \end{aligned}$$

  the direct limit running through all open compact subgroups $U = \prod _v U_v \subset G({\mathbb {A}}_{F^+}^\infty )$. Then ${\mathcal {A}}_{\varvec{\lambda }}$ is an semi-simple admissible $\overline{{\mathbb {Q}}}_p[G({\mathbb {A}}_{F^+}^\infty )]$-module. If $\iota {:}\,\overline{{\mathbb {Q}}}_p \cong {\mathbb {C}}$ is a fixed choice of isomorphism, then for each irreducible submodule $\pi \subset {\mathcal {A}},\,\pi \otimes _{\overline{{\mathbb {Q}}}_p, \iota } {\mathbb {C}}$ is the finite part of an automorphic representation of $G({\mathbb {A}}_{F^+})$, with an infinite part that can be described explicitly in terms of ${\varvec{\lambda }}$.

Theorem 4.1

Let $U = \prod _v U_v \subset G({\mathbb {A}}_{F^+}^\infty )$ be an open compact subgroup, and let T be a finite set of finite places of $F^+$ such that ${\mathbb {T}}^T_{\varvec{\lambda }}(U, {\mathcal {O}})$ is defined.

• (i) Let $\pi \subset {\mathcal {A}}_{\varvec{\lambda }}$ be an irreducible submodule such that $\pi ^U \ne 0$. Then there exists a continuous semi-simple representation $\rho _p(\pi ){:}\,G_F \rightarrow {{\mathrm{GL}}}_n(\overline{{\mathbb {Q}}}_p)$, satisfying the following conditions:

  • (a) $\rho _p(\pi )^c \cong \rho _p(\pi )^\vee \epsilon ^{1-n}$.

  • (b) If v is a finite place of $F^+$ such that $U_v$ is a hyperspecial maximal compact subgroup, and w is a place of F dividing v, then $\rho _p(\pi )|_{G_{F_w}}$ is unramified.

  • (c) If $v \in S_p$, then $\rho _p(\pi )|_{G_{F_{\widetilde{v}}}}$ is de Rham, and for each $\tau \in \widetilde{I}_p$, we have

    $$\begin{aligned} \mathrm {HT}_\tau (r_p(\pi )) = \{ \lambda _{\tau , 1} + (n-1), \lambda _{\tau , 2} + (n-2), \ldots , \lambda _{\tau , n} \}. \end{aligned}$$

  • (d) If v is a finite place of $F^+$ split in F as $v = w w^c$, then there is an isomorphism $\mathrm {WD}(\rho _p(\pi )|_{G_{F_w}})^{\text {F-ss}} \cong {{\mathrm{rec}}}_{F_w}^T(\pi _w \circ \iota _w^{-1})$. In particular, if $v \in S_p$ and $\pi _v$ is unramified, then $\rho _p(\pi )|_{G_{F_{\widetilde{v}}}}$ is crystalline.

  • (e) If $\rho _p(\pi )$ is irreducible, then the pair $(\rho _p(\pi ), \epsilon ^{1-n} \delta _{F{/}F^+}^n)$ is polarized.

• (ii) Let ${\mathfrak {m}}\subset {\mathbb {T}}^T_{\varvec{\lambda }}(U, {\mathcal {O}})$ be a maximal ideal with residue field k. Then there exists a continuous semi-simple representation $\overline{\rho }_{\mathfrak {m}}{:}\,G_F \rightarrow {{\mathrm{GL}}}_n({\mathbb {T}}_{\varvec{\lambda }}(U, {\mathcal {O}}){/}{\mathfrak {m}})$ satisfying the following conditions:

  • (a) $\overline{\rho }_{\mathfrak {m}}^c \cong \overline{\rho }_{\mathfrak {m}}^\vee \epsilon ^{1-n}$.

  • (b) If $v \not \in T$ is a finite place of $F^+$ and w is a place of F above v, then $\overline{\rho }_{\mathfrak {m}}|_{G_{F_w}}$ is unramified. If moreover w is split over $F^+$, then the characteristic polynomial of $\overline{\rho }_{\mathfrak {m}}({{\mathrm{Frob}}}_w)$ is equal to

    $$\begin{aligned} X^n + \cdots + (-1)^j q_w^{j(j-1){/}2} T_w^j X^{n-j} + \cdots + (-1)^n q_w^{n(n-1){/}2} T_w^n \in ({\mathbb {T}}_{\varvec{\lambda }}(U, {\mathcal {O}}){/}{\mathfrak {m}})[X]. \end{aligned}$$

  • (c) If $\overline{\rho }_{\mathfrak {m}}$ is irreducible, then the pair $(\overline{\rho }_{\mathfrak {m}}, \epsilon ^{1-n} \delta ^n_{F{/}F^+})$ is polarized.

• (iii) Let ${\mathfrak {m}}\subset {\mathbb {T}}^T_{\varvec{\lambda }}(U, {\mathcal {O}})$ be a maximal ideal with residue field k, and suppose that $\overline{\rho }_{\mathfrak {m}}$ is irreducible. Let $\overline{r}_{\mathfrak {m}}{:}\,G_{F^+} \rightarrow {\mathcal {G}}_n({\mathbb {T}}_{\varvec{\lambda }}(U, {\mathcal {O}}){/}{\mathfrak {m}})$ be a choice of extension of $\overline{\rho }_{\mathfrak {m}}$ (which exists, by Lemma 2.2, and satisfies $\nu \circ \overline{r}_{\mathfrak {m}}= \epsilon ^{1-n} \delta _{F{/}F^+}^n$, by the second part of the theorem). Then there exists a lifting $r_{\mathfrak {m}}{:}\,G_{F^+} \rightarrow {\mathcal {G}}_n({\mathbb {T}}_{\varvec{\lambda }}(U, {\mathcal {O}})_{\mathfrak {m}})$ of $\overline{r}_{\mathfrak {m}}$ satisfying the following conditions:

  • (a) $\nu \circ r_{\mathfrak {m}}= \epsilon ^{1-n} \delta _{F{/}F^+}^n$.

  • (b) If $v \not \in T$ is a finite place of $F^+$ and w is a place of F above v, then $r_{\mathfrak {m}}|_{G_{F_w}}$ is unramified. If moreover w is split over $F^+$, then the characteristic polynomial of $r_{\mathfrak {m}}|_{G_F}({{\mathrm{Frob}}}_w)$ is equal to

    $$\begin{aligned} X^n + \cdots + (-1)^j q_w^{j(j-1){/}2} T_w^j X^{n-j} + \cdots + (-1)^n q_w^{n(n-1){/}2} T_w^n \in {\mathbb {T}}^T_{\varvec{\lambda }}(U, {\mathcal {O}})_{\mathfrak {m}}[X]. \end{aligned}$$

    Moreover, $r_{\mathfrak {m}}$ is uniquely determined by these conditions, up to strict equivalence.

Proof

The first part follows by the argument in the proof of [14, Theorem 2.3], together with local-global compatibility for RACSDC automorphic representations of ${{\mathrm{GL}}}_n({\mathbb {A}}_F)$, in its strong form (see [6, 7]), and Theorem 3.2. The second part follows from the first part by reduction modulo p.

To prove the third part, we use the Galois representations constructed in the first. The algebra ${\mathbb {T}}^T_{\varvec{\lambda }}(U, {\mathcal {O}})_{\mathfrak {m}}$ is ${\mathcal {O}}$-flat and reduced, of dimension 1. For each minimal prime ${\mathfrak {p}}\subset {\mathbb {T}}^T_{\varvec{\lambda }}(U, {\mathcal {O}})_{\mathfrak {m}}$, we can find an ${\mathcal {O}}$-embedding ${\mathbb {T}}^T_{\varvec{\lambda }}(U, {\mathcal {O}})_{\mathfrak {m}}{/}{\mathfrak {p}}\hookrightarrow \overline{{\mathbb {Q}}}_p$ and an irreducible subrepresentation $\pi \subset {\mathcal {A}}_{\varvec{\lambda }}$ such that $\pi ^U \ne 0$ and for each unramified Hecke operator $T_w^i \in {\mathbb {T}}^T_{\varvec{\lambda }}(U, {\mathcal {O}})$, the image of $T_w^i$ in ${\mathbb {T}}^T_{\varvec{\lambda }}(U, {\mathcal {O}})_{\mathfrak {m}}{/}{\mathfrak {p}}\subset \overline{{\mathbb {Q}}}_p$ equals the eigenvalue of $T_w^i$ on $\pi ^U$.

In particular, we can find (by combining the first part of the lemma, Lemmas 2.2 and 2.3) a finite extension $K_{\mathfrak {p}}$ of ${\mathbb {T}}^T_{\varvec{\lambda }}(U, {\mathcal {O}})_{\mathfrak {m}}{/}{\mathfrak {p}}[1{/}p]$ inside $\overline{{\mathbb {Q}}}_p$ with ring of integers ${\mathcal {O}}_{K_{\mathfrak {p}}}$, together with a representation $r_{\mathfrak {p}}{:}\,G_{F^+} \rightarrow {\mathcal {G}}_n({\mathcal {O}}_{K_{\mathfrak {p}}})$ satisfying the following conditions:

• $r_{\mathfrak {p}}|_{G_{F}} \otimes _{{\mathcal {O}}_{K_{\mathfrak {p}}}} \overline{{\mathbb {Q}}}_p \cong r_p(\pi )$.

• If $v \not \in T$ is a finite place of $F^+$ and w is a place of F above v, then $r_{\mathfrak {p}}|_{G_{F_w}}$ is unramified. If moreover w is split over $F^+$, then the characteristic polynomial of $r_{\mathfrak {p}}|_{G_F}({{\mathrm{Frob}}}_w)$ is equal to

  $$\begin{aligned} X^n + \cdots + (-1)^j q_w^{j(j-1){/}2} T_w^j X^{n-j} + \cdots + (-1)^n q_w^{n(n-1){/}2} T_w^n \in ({\mathbb {T}}^T_{\varvec{\lambda }}(U, {\mathcal {O}})_{\mathfrak {m}}{/}{\mathfrak {p}})[X]. \end{aligned}$$

• We have $r_{\mathfrak {p}}\text { mod }{\mathfrak {m}}_{{\mathcal {O}}_{K_{\mathfrak {p}}}} = \overline{r}_{\mathfrak {m}}$ (i.e. equality, not just ${\mathcal {G}}_n^0(k)$-conjugacy).

Let ${\mathcal {O}}_{K_{\mathfrak {p}}}^0 \subset {\mathcal {O}}_{K_{\mathfrak {p}}}$ denote the subring consisting of elements whose image in the residue field ${\mathcal {O}}_{K_{\mathfrak {p}}}{/}{\mathfrak {m}}_{{\mathcal {O}}_{K_{\mathfrak {p}}}}$ in fact lies in k. Then $r_{\mathfrak {p}}$ is valued in ${\mathcal {G}}_n({\mathcal {O}}^0_{K_{\mathfrak {p}}})$, and we can form the lifting $r_{\mathfrak {m}}^0 = \times _{\mathfrak {p}}r_{\mathfrak {p}}$, valued in the ring

$$\begin{aligned} A = \times _{\mathfrak {p}}{\mathcal {O}}_{K_{\mathfrak {p}}}^0 = \left\{ (x_{\mathfrak {p}})_{\mathfrak {p}}\in \prod _{\mathfrak {p}}{\mathcal {O}}_{K_{\mathfrak {p}}}^0 \mid x_{\mathfrak {p}}\text { mod }{\mathfrak {m}}_{{\mathcal {O}}_{K_{\mathfrak {p}}}} \text { independent of }{\mathfrak {p}}\right\} . \end{aligned}$$

(thus the pullback is taken relative to k). There is a natural embedding ${\mathbb {T}}_{\varvec{\lambda }}^T(U, {\mathcal {O}})_{\mathfrak {m}}\hookrightarrow A$, and for each $\delta \in G_F$, we have ${{\mathrm{tr}}}r_{\mathfrak {m}}^0|_{G_F}(\delta ) \in {\mathbb {T}}_{\varvec{\lambda }}^T(U, {\mathcal {O}})_{\mathfrak {m}}$. Indeed, by the Chebotarev density theorem it suffices to check that ${{\mathrm{tr}}}r_{\mathfrak {m}}^0|_{G_F}({{\mathrm{Frob}}}_w)$ lies in ${\mathbb {T}}_{\varvec{\lambda }}^T(U, {\mathcal {O}})_{\mathfrak {m}}$, when w is a finite place of F split over $F^+$ and not dividing T; but this follows from our knowledge of the characteristic polynomial of $r_{\mathfrak {m}}^0|_{G_F}({{\mathrm{Frob}}}_w)$.

By Lemma 2.4, therefore, we can find a ${\mathcal {G}}_n^0(A)$-conjugate $r_{\mathfrak {m}}$ of $r_{\mathfrak {m}}^0$ which is valued in ${\mathcal {G}}_n({\mathbb {T}}_{\varvec{\lambda }}^T(U, {\mathcal {O}})_{\mathfrak {m}})$. Then $r_{\mathfrak {m}}$ is a lifting of $\overline{r}_{\mathfrak {m}}$ which clearly has the desired properties. By Lemma 2.5, $r_{\mathfrak {m}}$ is uniquely determined by these properties, up to strict equivalence. This completes the proof. $\square$

We now specialize to our situation of interest. Fix a finite set T of finite places of $F^+$, split in F, and containing $S_p$, and let $U = \prod _v U_v \subset G({\mathbb {A}}_{F^+}^\infty )$ be an open compact subgroup satisfying the following conditions:

• If $v \in S_p$, then $U_v = \iota _{\widetilde{v}}^{-1} {{\mathrm{GL}}}_n({\mathcal {O}}_{F_{\widetilde{v}}})$.

• If $v \not \in T$ is a finite place of $U_v$, then $U_v$ is a hyperspecial maximal compact subgroup of $G(F_v^+)$.

• For all $g \in G({\mathbb {A}}_{F^+}^\infty )$, the (a priori finite) group $U \cap g G(F^+) g^{-1}$ is trivial.

Fix a choice of ${\varvec{\lambda }}\in ({\mathbb {Z}}_+^n)^{\widetilde{I}_p}$, and let $\pi \subset {\mathcal {A}}_{\varvec{\lambda }}$ be an irreducible submodule such that $\pi ^U \ne 0$ and $\overline{\rho _p(\pi )}$ is irreducible. Let ${\mathfrak {m}}\subset {\mathbb {T}}_{\varvec{\lambda }}^T(U, {\mathcal {O}})$ be the maximal ideal associated to the Hecke eigenvalues of $\pi$. Then there is an isomorphism $\overline{r_p(\pi )} \cong \overline{\rho }_{\mathfrak {m}}$. We fix a choice of extension $\overline{r}_{\mathfrak {m}}$ and lifting $r_{\mathfrak {m}}$, as in the statement of Theorem 4.1. Let $S = T \cup S_\infty$, and consider the global deformation problem

$$\begin{aligned} {\mathcal {S}}= \left( F, \overline{r}_{\mathfrak {m}}, {\mathcal {O}}, \epsilon ^{1-n} \delta _{F{/}F^+}^n, S, \{ {\mathcal {D}}_v^{ {\varvec{\lambda }}_v,\text {cr}} \}_{v \in S_p} \cup \{ {\mathcal {D}}_v^{ \text {fl}} \}_{v \in T - S_p}\right) . \end{aligned}$$

(the local deformation problems have been defined in Sect. 2.3). It follows immediately from the construction of Theorem 4.1 that the lifting $r_{\mathfrak {m}}$ is of type ${\mathcal {S}}$, so is classified by a surjective homomorphism $R_{\mathcal {S}}\rightarrow {\mathbb {T}}_{\varvec{\lambda }}^T(U, {\mathcal {O}})_{\mathfrak {m}}$.

Theorem 4.2

Let $f{:}\,R_{\mathcal {S}}\rightarrow {\mathcal {O}}$ be a homomorphism corresponding to a lifting $r{:}\,G_{F^+} \rightarrow {\mathcal {G}}_n({\mathcal {O}})$ of $\overline{r}_{\mathfrak {m}}$. Suppose that:

• (i) The subgroup $\overline{\rho }_{\mathfrak {m}}(G_F) \subset {{\mathrm{GL}}}_n(k)$ is adequate, in the sense of Definition 2.20.

• (ii) For each $v \in T$, we have $\rho |_{G_{F^+_v}} \sim r_p(\pi )|_{G_{F_v^+}}$.

• (iii) If p is odd, then $F = F^+(\zeta _p)$. If $p = 2$, then $F = F^+(\sqrt{-1})$.

• (iv) If $p = 2$ and n is even, then there exists a place $v | \infty$ such that $(\overline{\rho }_{\mathfrak {m}}, \epsilon ^{1-n} \delta _{F{/}F^+}^n)$ is strongly residually odd, in the sense of Definition 3.3.

Then there exists an irreducible submodule $\sigma \subset {\mathcal {A}}_{\varvec{\lambda }}$ such that $\sigma ^U \ne 0$ and $\rho _p(\sigma ) \cong r|_{G_F}$.

Proof

The proof is essentially the same as the proof of [32, Theorem 6.8], with references to Proposition 4.4 of op. cit. replaced with references to Proposition 2.21 of this article. We therefore feel free to only sketch parts of the proof that are substantially the same as the proof of [32, Theorem 6.8]. After possibly enlarging ${\mathcal {O}}$, we can find a homomorphism $f'{:}\,{\mathbb {T}}_{\varvec{\lambda }}^T(U, {\mathcal {O}})_{\mathfrak {m}}\rightarrow {\mathcal {O}}$ such that $(f' \circ r_{\mathfrak {m}})|_{G_{F}} \otimes _{\mathcal {O}}\overline{{\mathbb {Q}}}_p \cong \rho _p(\pi )$.

Let $q = h^1_{{\mathcal {S}}^\perp , T} - 1$ and $g = q + |T| - 1 - [F^+{:}\,{\mathbb {Q}}]n(n-1){/}2$. By Proposition 2.21, we can find for each $N \ge 1$ a Taylor–Wiles datum $(Q_N, ( \alpha _v )_{v \in Q})$ of level N such that $\# Q_N = q$ and the map $R_{{\mathcal {S}}, T}^\text {loc} \rightarrow R_{{\mathcal {S}}_{Q_N}}^T$ can be extended to a surjection $R_{{\mathcal {S}}, T}^\text {loc}\llbracket X_1, \ldots , X_{g} \rrbracket \twoheadrightarrow R_{{\mathcal {S}}_{Q_N}}^T$ (we write here ${\mathcal {S}}_{Q_N}$ for the auxiliary deformation problem constructed in Sect. 2.4. We remark that in applying Proposition 2.21, we are using hypotheses (i), and (iii) and (iv) of the theorem). We define $\Delta _{Q_N} = \prod _{v \in Q_N} k(v)^\times (p)$, as in Sect. 2.4. Then $R_{{\mathcal {S}}_{Q_N}}$ is an ${\mathcal {O}}[\Delta _{Q_N}]$-algebra, and there is a canonical isomorphism $R_{{\mathcal {S}}_{Q_N}} \otimes _{{\mathcal {O}}[\Delta _{Q_N}]} {\mathcal {O}}\cong R_{{\mathcal {S}}}$.

Let $H = S_{\varvec{\lambda }}(U, {\mathcal {O}})_{\mathfrak {m}}$. Then H is a faithful ${\mathbb {T}}_{\varvec{\lambda }}(U, {\mathcal {O}})_{\mathfrak {m}}$-module, and becomes an $R_{{\mathcal {S}}}$-module via the surjective homomorphism $R_{\mathcal {S}}\rightarrow {\mathbb {T}}_{\varvec{\lambda }}(U, {\mathcal {O}})_{\mathfrak {m}}$ constructed above. Arguing exactly as in [32, § 5] and the proof of [32, Theorem 6.8], we can construct for each $N \ge 1$ an $R_{{\mathcal {S}}_{Q_N}}$-module $H_{Q_N}$, free over ${\mathcal {O}}[\Delta _{Q_N}]$, together with an isomorphism $H_{Q_N} \otimes _{{\mathcal {O}}[\Delta _{Q_N}]} {\mathcal {O}}\cong H$ of $R_{{\mathcal {S}}_{Q_N}}$-modules (we remark that it is assumed at some point in [32, § 5] that the residue characteristic is odd; however, this assumption plays no role in the proof).

Let ${\mathcal {T}}= {\mathcal {O}}\llbracket X_1, \ldots , X_{(n^2 + 1)|T| - 1} \rrbracket$. Choose a representative $r_{\mathcal {S}}{:}\,G_{F^+} \rightarrow {\mathcal {G}}_n(R_{\mathcal {S}})$ of the universal deformation, and for each $N \ge 1$ a representative $r_{{\mathcal {S}}_{Q_N}}{:}\,G_{F^+} \rightarrow {\mathcal {G}}_n(R_{{\mathcal {S}}_{Q_N}})$ lifting $R_{\mathcal {S}}$. These choices determine canonical isomorphisms $R_{\mathcal {S}}^T \cong R_{\mathcal {S}}\widehat{\otimes }_{\mathcal {O}}{\mathcal {T}}$ and $R_{{\mathcal {S}}_{Q_N}}^T \cong R_{{\mathcal {S}}_{Q_N}} \widehat{\otimes }_{\mathcal {O}}{\mathcal {T}}$, and give $H^T = H \widehat{\otimes }_{\mathcal {O}}{\mathcal {T}}$ and $H^T_{Q_N} = H_{Q_N} \widehat{\otimes }_{\mathcal {O}}{\mathcal {T}}$ the structure of $R_{\mathcal {S}}^T$ and $R_{{\mathcal {S}}_{Q_N}}^T$-module, respectively.

We choose for each $N \ge 1$ a surjection $R_{{\mathcal {S}}, T}^\text {loc}\llbracket X_1, \ldots , X_{g} \rrbracket \twoheadrightarrow R_{{\mathcal {S}}_{Q_N}}^T$. We let $R_\infty = R_{{\mathcal {S}}, T}^\text {loc}\llbracket X_1, \ldots , X_{g} \rrbracket$ and $S_\infty = {\mathcal {O}}\llbracket {\mathbb {Z}}_p^q \rrbracket \widehat{\otimes }_{\mathcal {O}}{\mathcal {T}}$, and choose for each $N \ge 1$ a surjection ${\mathbb {Z}}_p^q \rightarrow \Delta _{Q_N}$ (which gives rise to a surjection $S_\infty \rightarrow {\mathcal {T}}[ \Delta _{Q_N} ]$).

After patching exactly as in the proof of [32, Theorem 6.8], we obtain the following objects:

• A ring homomorphism $S_\infty \rightarrow R_\infty$, together with an $R_\infty$-module $H_\infty$, free over $S_\infty$.

• A surjection $R_\infty \rightarrow R_{\mathcal {S}}$ of $S_\infty$-algebras, together with an isomorphism $H_\infty \otimes _{S_\infty } {\mathcal {O}}\cong H$ of $R_\infty$-modules.

In particular, we have

$$\begin{aligned}&{{\mathrm{depth}}}_{R_\infty } H_\infty \ge {{\mathrm{depth}}}_{S_\infty } H_\infty = \dim S_\infty = 1 + q +( n^2 + 1 )|T|- 1 \\&\quad = q + (n^2 + 1)|T| = \dim R_\infty . \end{aligned}$$

It follows that ${{\mathrm{depth}}}_{R_\infty } H_\infty = \dim R_\infty$, and ${{\mathrm{Supp}}}_{R_\infty } H_\infty \subset {{\mathrm{Spec}}}R_\infty$ is a union of irreducible components, by [31, Lemma 2.3]. For each $v \in T$, let ${\mathcal {C}}_v$ be the unique irreducible component of ${{\mathrm{Spec}}}R_v^{{\varvec{\lambda }}_v, \text {cr}}$ or ${{\mathrm{Spec}}}R_v^{\text {fl}}$ containing the point corresponding to $r_p(\pi )$ (if $v \in S_p$, it is unique because $R_v^{{\varvec{\lambda }}_v, \text {cr}}[1{/}p]$ is formally smooth over E. If $v \in T - S_p$, it is unique because $\pi _v$ is generic, cf. Sect. 2.3.4). Let $R_v^{{\mathcal {C}}_v}$ denote the affine ring of ${\mathcal {C}}_v$, endowed with its reduced subscheme structure, and let $R_\infty ' = (\widehat{\otimes }_{v \in T} R_v^{{\mathcal {C}}_v})\llbracket X_1, \ldots , X_g \rrbracket$. Then ${{\mathrm{Spec}}}R'_\infty \subset {{\mathrm{Spec}}}R_\infty$ is an irreducible component, which lies in the support of $H_\infty$ as $R_\infty$-module; indeed, the closed point $x'$ of ${{\mathrm{Spec}}}R_\infty [1{/}p]$ corresponding to the composite

$$\begin{aligned} R_\infty \rightarrow R_{\mathcal {S}}\rightarrow {\mathbb {T}}^T_{\varvec{\lambda }}(U, {\mathcal {O}})_{\mathfrak {m}}\rightarrow {\mathcal {O}}\end{aligned}$$

given by $f'$ lies in ${{\mathrm{Spec}}}R'_\infty [1{/}p]$, and ${{\mathrm{Spec}}}R'_\infty [1{/}p]$ is (by hypothesis 2 of the theorem) the unique irreducible component of ${{\mathrm{Spec}}}R_\infty [1{/}p]$ containing this closed point. It follows that the closed point x of ${{\mathrm{Spec}}}R_\infty [1{/}p]$ corresponding to f is also in the support of $H_\infty$, and hence that

$$\begin{aligned} H_\infty \otimes _{R_\infty , f} {\mathcal {O}}[1{/}p] \cong H \otimes _{R_{\mathcal {S}}, f} {\mathcal {O}}[1{/}p] \ne 0. \end{aligned}$$

This is only possible if f factors through the quotient $R_{\mathcal {S}}\rightarrow {\mathbb {T}}_{\varvec{\lambda }}^T(U, {\mathcal {O}})_{\mathfrak {m}}$ or, equivalently, if there exists an irreducible submodule $\sigma \subset {\mathcal {A}}_{\varvec{\lambda }}$ such that $\sigma ^U \ne 0$ and $r|_{G_F} \otimes _{\mathcal {O}}\overline{{\mathbb {Q}}}_p \cong \rho _p(\sigma )$. This completes the proof. $\square$

We now state as a corollary a consequence of Theorem 4.2 for Galois representations valued in ${{\mathrm{GL}}}_n$.

Corollary 4.3

Let F be an imaginary CM number field with totally real subfield $F^+$, and let T be a finite set of finite places of $F^+$ containing $S_p$ and split in F. We assume that F satisfies the following conditions:

• (i) The extension $F{/}F^+$ is everywhere unramified, and $n [ F^+{:}\,{\mathbb {Q}}] \equiv 0 \text { mod }4$.

• (ii) If p is odd, then $F = F^+(\zeta _p)$. If $p = 2$, then $F = F^+(\sqrt{-1})$.

Let $\rho {:}\,G_F \rightarrow {{\mathrm{GL}}}_n(\overline{{\mathbb {Q}}}_p)$ be a continuous representation satisfying the following conditions:

• (i) The group $\overline{\rho }(G_F)$ is adequate. In particular, $\overline{\rho }$ is absolutely irreducible.

• (ii) We have $\rho ^c \cong \rho ^\vee \epsilon ^{1-n}$.

• (iii) If w is a finite place of F not dividing T, then $\rho |_{G_{F_w}}$ is unramified.

• (iv) If w is a finite place of F dividing p, then $\rho |_{G_{F_w}}$ is crystalline.

• (v) There exists a RACSDC automorphic representation $\Pi$ of weight ${\varvec{\lambda }}$ and an isomorphism $\iota {:}\,\overline{{\mathbb {Q}}}_p \cong {\mathbb {C}}$ satisfying the following conditions:

  • (a) There is an isomorphism $\overline{r_\iota (\Pi )} \cong \overline{\rho }$.

  • (b) If w is a finite place of F not dividing T, then $\Pi _w$ is unramified.

  • (c) If w is a finite place of F dividing p, then $\Pi _w$ is unramified.

  • (d) If w is a finite place of F dividing T, then $r_\iota (\Pi )|_{G_{F_w}} \sim \rho |_{G_{F_w}}$.

  • (e) If $p = 2$ and n is even, then there exists a place $v|\infty$ such that $\overline{\rho }$ is strongly residually odd at v.

Then $\rho$ is automorphic: there exists a RACSDC automorphic representation $\Sigma$ of weight ${\varvec{\lambda }}$ and an isomorphism $\rho \cong r_\iota (\Sigma )$.

Proof

After possibly enlarging T, we can assume that it contains a place $v_0$, prime to p, such that both $\Pi$ and $\rho$ are unramified above $v_0$. We introduce a unitary group G as at the beginning of Sect. 4. By [22, Théorème 5.4], there exists an isomorphism $\iota {:}\,\overline{{\mathbb {Q}}}_p \cong {\mathbb {C}}$ and an irreducible submodule $\pi \subset {\mathcal {A}}_{\varvec{\lambda }}$ satisfying the following conditions:

• If v is an inert place of $F^+$, then $\pi _v$ is unramified.

• If $v = ww^c$ is a split place of $F^+$, then there is an isomorphism $\pi _v \cong \iota ^{-1}\Pi _w \circ \iota _w^{-1}$.

Then $\rho _p(\pi ) \cong r_\iota (\Pi )$. By Lemmas 2.3 and 2.2, together with the fact that $\overline{\rho _p(\pi )} \cong \overline{\rho }$, we can find a coefficient field E and a homomorphism $r{:}\,G_{F^+} \rightarrow {\mathcal {G}}_n({\mathcal {O}})$ satisfying the following conditions:

• $r|_{G_F} \otimes _{\mathcal {O}}\overline{{\mathbb {Q}}}_p \cong \rho$.

• $\nu \circ r = \epsilon ^{1-n} \delta _{F{/}F^+}^n$.

Let $U = \prod _v U_v \subset G({\mathbb {A}}_{F^+}^\infty )$ be the open compact subgroup defined as follows:

• If $v \not \in T$, then $U_v$ is a hyperspecial maximal compact subgroup of $G(F_v^+)$.

• If $v \in S_p$, then $U_v = \iota _{{\widetilde{v}}}^{-1}({{\mathrm{GL}}}_n({\mathcal {O}}_{F_{\widetilde{v}}}))$.

• If $v \in T - S_p$, then $U_v$ is any torsion-free subgroup of $G(F_v^+)$ such that $\pi _v^{U_v} \ne 0$.

Then $\pi ^U \ne 0$, and for all $g \in G({\mathbb {A}}_{F^+}^\infty )$, the group $U \cap g G(F^+) g^{-1}$ is trivial. We can then apply Theorem 4.2 to r, to deduce the existence of an irreducible subrepresentation $\sigma \subset {\mathcal {A}}_{\varvec{\lambda }}$ such that $\rho \cong r_p(\sigma )$. We then deduce from [22, Corollaire 5.3] the existence of a RACDSC automorphic representation $\Sigma$ of ${{\mathrm{GL}}}_n({\mathbb {A}}_F)$ such that $\rho \cong r_\iota (\Sigma )$. This completes the proof. $\square$

5 An automorphy lifting theorem

Theorem 5.1

Let $n \ge 2$. Let F be an imaginary CM number field with totally real subfield $F^+$. Fix a prime p and an isomorphism $\iota {:}\,\overline{{\mathbb {Q}}}_p \cong {\mathbb {C}}$, and consider a continuous representation

$$\begin{aligned} \rho {:}\,G_F \rightarrow {{\mathrm{GL}}}_n(\overline{{\mathbb {Q}}}_p). \end{aligned}$$

Suppose that $\rho$ satisfies the following conditions:

• (i) There is an isomorphism $\rho ^c \cong \rho ^\vee \epsilon ^{1-n}$.

• (ii) The group $\overline{\rho }(G_{F(\zeta _p)}) \subset {{\mathrm{GL}}}_n(\overline{{\mathbb {F}}}_p)$ is adequate, in the sense of Definition 2.20.

• (iii) The representation $\rho$ is almost everywhere unramified.

• (iv) There exists a RACSDC automorphic representation $\pi$ of ${{\mathrm{GL}}}_n({\mathbb {A}}_F)$ such that:

  • (a) There is an isomorphism $\overline{r_\iota (\pi )} \cong \overline{\rho }$.

  • (b) For each finite place v of F, we have $r_\iota (\pi )|_{G_{F_v}} \sim \rho |_{G_{F_v}}$ (this condition is automatic if $\pi _v$ and $\rho |_{G_{F_v}}$ are both unramified). In particular, if v | p, then $\rho |_{G_{F_v}}$ and $r_\iota (\pi )|_{G_{F_v}}$ are potentially crystalline.

• (v) If $p = 2$ and n is even, then there exists a place $v | \infty$ of $F^+$ at which the pair $(\overline{\rho }, \epsilon ^{1-n} \delta _{F{/}F^+}^n)$ is strongly residually odd, in the sense of Definition 3.3.

Then $\rho$ is automorphic: there exists a RACSDC automorphic representation $\Pi$ of ${{\mathrm{GL}}}_n({\mathbb {A}}_F)$ such that $\rho \cong r_\iota (\Pi )$.

Remark

Before giving the proof of Theorem 5.1, we compare it with [32, Theorem 7.1], in the case that the character $\mu$ in loc. cit. is taken to be trivial. Then the hypotheses and conclusions of the two theorems are essentially the same, except for the following:

• (i) Here we do not assume that the residue characteristic p is odd (the main point of this paper).

• (ii) We do not assume that $\zeta _p \not \in F$.

• (iii) The definition of ‘adequate subgroup’ used here (i.e. Definition 2.20) is more general than that of [32, Definition 2.3] in the case that p is odd and p | n.

Proof of Theorem 5.1

We reduce the theorem to Corollary 4.3, using the technique of soluble base change (i.e. using [5, Lemma 2.2.2]). First, we can replace F by $F(\zeta _p)$ (if p is odd) or $F(\sqrt{-1})$ (if $p = 2$). If $p = 2$ and n is even, then it is necessary to check that this preserves the condition that $(\overline{\rho }, \epsilon ^{1-n} \delta _{F{/}F^+}^n)$ is strongly residually odd at some infinite place, but this follows easily from the definitions.

It now suffices to observe that we can find a soluble extension L / F satisfying the following conditions:

• $n [ L^+{:}\,{\mathbb {Q}}] \equiv 0 \text { mod }4$ and $L{/}L^+$ is everywhere unramified.

• $\overline{\rho }(G_L) = \overline{\rho }(G_F)$.

• If v is a place of L dividing p, then $\rho |_{G_{F_v}}$ is crystalline and v is split over $L^+$.

• If v is a finite place of L not dividing p, then $\rho |_{G_{F_v}}$ is unipotently ramified and $\pi _v$ has an Iwahori-fixed vector. If either of $\rho |_{G_{F_v}}$ or $\pi _v$ is ramified, then v is split over $L^+$.

We can now apply Corollary 4.3 to deduce that $\rho |_{G_L}$ is automorphic, and hence that $\rho$ is automorphic. This concludes the proof. $\square$

6 Application: the case of ${{\mathrm{GL}}}_2$

In this section, we apply Theorem 5.1 to improve a modularity lifting theorem of Kisin [18] for 2-adic potentially Barsotti-Tate representations. More precisely, we remove the local conditions at p but add a local condition at $\infty$. The theorem we prove is the following:

Theorem 6.1

Let F be a totally real number field, and let $\rho {:}\,G_F \rightarrow {{\mathrm{GL}}}_2(\overline{{\mathbb {Q}}}_2)$ be a continuous representation satisfying the following conditions:

• (i) $\rho$ is almost everywhere unramified.

• (ii) For each place v|2 of $F,\,\rho |_{G_{F_v}}$ is potentially crystalline. For each embedding $\tau {:}\,F_v \hookrightarrow \overline{{\mathbb {Q}}}_p,\,\mathrm {HT}_\tau (\rho ) = \{ 0, 1 \}$.

• (iii) $\overline{\rho }$ is absolutely irreducible, and has non-soluble image. There exists a place $v|\infty$ of F such that $\overline{\rho }(c_v)$ is non-trivial.

• (iv) There exists a RAESDC automorphic representation $(\pi , \chi )$ of ${{\mathrm{GL}}}_2({\mathbb {A}}_F)$ and an isomorphism $\iota {:}\,\overline{{\mathbb {Q}}}_2 \cong {\mathbb {C}}$ such that $\overline{r_\iota (\pi )} \cong \overline{\rho }$.

Then $\rho$ is automorphic: there exists a RAESDC automorphic representation $(\sigma , \psi )$ of ${{\mathrm{GL}}}_2({\mathbb {A}}_F)$ such that $\rho \cong r_\iota (\sigma )$.

(We remind the reader that the condition that $\overline{\rho }$ has non-soluble image is equivalent, by the classification of finite subgroups of ${\mathrm {PGL}}_2(\overline{{\mathbb {F}}}_2)$, to asking that the projective image of $\overline{\rho }(G_F)$ is a conjugate of ${\mathrm {PSL}}_2({\mathbb {F}}_{2^a})$ for some $a > 1$. In particular, the projective image is a simple group.) By [18, Proposition 3.2.9], to prove Theorem 6.1, it is enough to show the following result:

Theorem 6.2

Let F be a totally real number field, and let $(\pi , \chi )$ be a RAESDC automorphic representation of ${{\mathrm{GL}}}_2({\mathbb {A}}_F)$. Fix an isomorphism $\iota {:}\,\overline{{\mathbb {Q}}}_2 \cong {\mathbb {C}}$. Suppose that $\overline{r_\iota (\pi )}$ is absolutely irreducible, with non-soluble image, and that there exists a place $v | \infty$ of F such that $\overline{r_\iota (\pi )}(c_v)$ is non-trivial. Then there exists a soluble totally real extension $F'{/}F$ and a RAESDC automorphic representation $(\sigma , \psi )$ of ${{\mathrm{GL}}}_2({\mathbb {A}}_{F'})$ satisfying the following conditions:

• (i) $\sigma$ has weight 0 and $\overline{r_\iota (\sigma )} \cong \overline{r_\iota (\pi )}|_{G_{F'}}$.

• (ii) For each place v|2 of $F',\,\sigma _v$ is unramified and $\iota$-ordinary.

The idea of proving a result like Theorem 6.2, and applying it to improve Kisin’s results is due to Barnet-Lamb, Gee, and Geraghty, and has been carried out by them in the case that the residue characteristic p is odd [3, 4]. Here we will use our main theorem to generalize their technique to the case $p = 2$.

The main idea, as in [3, Introduction], is to construct lifts with prescribed local properties by using tensor products. However, we cannot use automorphic induction from ${{\mathrm{GL}}}_2$ to ${{\mathrm{GL}}}_4$, as the induction of a representation from an even degree extension can never be adequate in characteristic 2. Instead, we use the ${{\mathrm{GL}}}_2 \times {{\mathrm{GL}}}_2 \rightarrow {{\mathrm{GL}}}_4$ tensor product functoriality, and check that adequacy is preserved in our case of interest.

We begin with some preliminary results about tensor products.

Lemma 6.3

Let F be a number field, and let k be a finite field of characteristic 2. Let $r_1, r_2{:}\,G_F \rightarrow {{\mathrm{GL}}}_2(k)$ be absolutely irreducible representations with non-soluble image, and suppose that the extensions $F({{\mathrm{ad}}}r_1)$ and $F({{\mathrm{ad}}}r_2)$ are disjoint over F. Then the group $(r_1 \otimes r_2)(G_F) \subset {{\mathrm{GL}}}_4(k)$ is adequate.

Proof

Let $H_1 = r_1(G_F),\,H_2 = r_2(G_F)$, and $r = r_1 \otimes r_2$ and $H = r(G_F) \subset {{\mathrm{GL}}}_4(k)$. We recall that we must show the following:

• We have $H^1(H, k) = 0$ and $H^1(H, {{\mathrm{ad}}}_0 r) = 0$.

• For each simple k[H]-submodule $W \subset {{\mathrm{ad}}}r$, there exists a semi-simple element $\sigma \in H$ with an eigenvalue $\alpha \in K$ such that ${{\mathrm{tr}}}e_{\sigma , \alpha } W \ne 0$.

We first note that the groups $H_1, H_2$ are adequate. Indeed, they have no 2-power quotients, and the group $H^1(H, {{\mathrm{ad}}}_0 r)$ is 0, by [10, Lemma 42]. The unique simple $k[H_1]$-submodule of ${{\mathrm{ad}}}r_1$ is the subspace $Z_1$ of scalar matrices, and we have ${{\mathrm{tr}}}e_{\sigma , \alpha } Z_1 \ne 0$ for any element $\sigma \in H$ with two distinct eigenvalues $\alpha , \beta \in k$.

It follows immediately that the group H has no 2-power quotients, since it is itself a quotient of $H_1 \times H_2$. To show that $H^1(H, {{\mathrm{ad}}}_0 r) = 0$, it is enough (by inflation-restriction) to show that $H^1(H_1 \times H_2, {{\mathrm{ad}}}_0 r) = 0$. By the Künneth formula, we have

$$\begin{aligned} H^1(H_1 \times H_2, {{\mathrm{ad}}}r) = H^1(H_1 \times H_2, {{\mathrm{ad}}}r_1 \otimes _k {{\mathrm{ad}}}r_2) = 0, \end{aligned}$$

so it is enough to show that the map $H^2(H_1 \times H_2, k) \rightarrow H^2(H_1 \times H_2, {{\mathrm{ad}}}r)$ arising from the short exact sequence

is injective. However, this map is identified under the Künneth isomorphism with a map

$$\begin{aligned} H^2(H_1 \times H_2, k)\cong & {} H^2(H_2, k) \oplus H^2(H_1, k) \rightarrow H^2(H_2, {{\mathrm{ad}}}r_2) \oplus H^2(H_1, {{\mathrm{ad}}}r_1)\\\cong & {} H^2(H_1 \times H_2, {{\mathrm{ad}}}r), \end{aligned}$$

since $H^1(H_1, k) = H^1(H_2, k) = 0$. It is therefore a sum of the maps $H^2(H_i, k) \rightarrow H^2(H_i, {{\mathrm{ad}}}r_i)$, which are injective because the groups $H_i$ are adequate.

The space ${{\mathrm{ad}}}r$ has a unique simple submodule $Z_1 \otimes _k Z_2 = Z$, the subspace of scalar matrices. To establish the second point in the definition of adequacy, it is therefore enough to show that H contains a semi-simple element $\sigma$ with an eigenvalue of multiplicity 1. Since the projective images of $H_1$ and $H_2$ are assumed equal to conjugates of ${\mathrm {PGL}}_2({\mathbb {F}}_{2^{a_i}})$ for some $a_i > 1$, it is easy to see that this can be achieved. $\square$

Lemma 6.4

Let $F'{/}F$ be a Galois extension of totally real number fields, let $(\pi , \chi )$ be a RAESDC automorphic representation of ${{\mathrm{GL}}}_n({\mathbb {A}}_{F'})\,(n \ge 2)$, let p be a prime, and let $\iota {:}\,\overline{{\mathbb {Q}}}_p \cong {\mathbb {C}}$ be an isomorphism such that $r_\iota (\pi )$ is irreducible. Let $\rho {:}\,G_F \rightarrow {{\mathrm{GL}}}_n(\overline{{\mathbb {Q}}}_p)$ be a continuous representation such that $\rho |_{G_{F'}} \cong r_\iota (\pi )$. Finally, let S be a finite set of places of F, containing the infinite places, the places above p, and any place above which $\rho$ or $F'$ is ramified. Then the L-function

$$\begin{aligned} L^S(\iota \rho , s) = \prod _{v \not \in S} \det (1 - \iota \rho ({{\mathrm{Frob}}}_v) q_v^{-s})^{-1} \end{aligned}$$

has a meromorphic continuation to ${\mathbb {C}}$ and is analytic and non-vanishing at the point $s=\frac{1}{2}(1 + n - \alpha )$, where $\alpha \in {\mathbb {Z}}$ is the unique integer such that $\chi \Vert \cdot \Vert _{F'}^{-\alpha }$ has finite order.

Proof

By Brauer’s theorem, we can find subfields $F'{/}F_i{/}F,\,i = 1, \ldots , m$, with ${{\mathrm{Gal}}}(F'{/}F_i)$ soluble, integers $n_i$, and finite order characters $\psi _i{:}\,G_{F_i} \rightarrow \overline{{\mathbb {Q}}}_p^\times$, such that

$$\begin{aligned} \rho = \sum _i n_i {{\mathrm{Ind}}}_{G_F}^{G_{F_i}} \left( \rho |_{G_{F_i}} \otimes \psi _i\right) \end{aligned}$$

in the Grothendieck group of continuous representations of $G_F$. Let $S_i$ denote the set of places of $F_i$ above S; we then have a similar identity of L-functions (valid a priori in a right half-plane where the relevant Euler products converge absolutely)

$$\begin{aligned} L^S(\iota \rho , s) = \prod _{i=1}^m L^S\left( {{\mathrm{Ind}}}_{G_F}^{G_{F_i}}( \iota \rho |_{G_{F_i}} \otimes \psi _i), s\right) ^{n_i} = \prod _{i=1}^m L^{S_i}\left( \iota \rho |_{G_{F_i}} \otimes \psi _i, s\right) ^{n_i}. \end{aligned}$$

For each $i = 1, \ldots , m$, the representation $\pi$ descends to a cuspidal automorphic representation $\pi _i$ of ${{\mathrm{GL}}}_n({\mathbb {A}}_{F_i})$. We can therefore rewrite the above product as

$$\begin{aligned}&L^S(\iota \rho , s) = \prod _{i=1}^m L^{S_i}( \pi _i \Vert \cdot \Vert _{F_i}^{(1-n){/}2}\otimes \iota \psi _i, s)^{n_i} \nonumber \\&\quad = \prod _{i=1}^m L^{S_i}\left( \pi _i \Vert \cdot \Vert _{F_i}^{-\alpha {/} 2}\otimes \iota \psi _i, s + \frac{1}{2}(\alpha + 1 - n)\right) ^{n_i}. \end{aligned}$$

Each $\pi _i \Vert \cdot \Vert _{F_i}^{-\alpha {/}2}\otimes \iota \psi _i$ is a unitary cuspidal automorphic representation of ${{\mathrm{GL}}}_n({\mathbb {A}}_{F_i})$. It is known (see [16, Theorem 1.3]) that each of the L-functions $L^{S_i}(\pi _i \Vert \cdot \Vert _{F_i}^{-\alpha {/}2}\otimes \iota \psi _i, s)$ has a meromorphic continuation to the whole complex plane, and is holomorphic and non-vanishing at the point $s = 1$. It follows that $L^S(\iota \rho , s)$ has a meromorphic continuation to the whole complex plane, and is holomorphic and non-vanishing at the point $s = \frac{1}{2}(1 + n - \alpha )$. This completes the proof. $\square$

Proposition 6.5

Let F be a totally real number field, let p be a prime, and let $\rho _1, \rho _2{:}\,G_F \rightarrow {{\mathrm{GL}}}_2(\overline{{\mathbb {Q}}}_p)$ be continuous representations such that $\rho _1 \otimes \rho _2,\,{{\mathrm{Sym}}}^2 \rho _1$ and ${{\mathrm{Sym}}}^2 \rho _2$ are irreducible. Suppose moreover that each of $\rho _1$ and $\rho _2$ is unramified outside finitely many places, and de Rham with distinct Hodge–Tate weights. Fix an isomorphism $\iota {:}\,\overline{{\mathbb {Q}}}_p \cong {\mathbb {C}}$.

• (i) Suppose there exist RAESDC automorphic representations $(\pi _1, \chi _1)$ and $(\pi _2, \chi _2)$ of ${{\mathrm{GL}}}_2({\mathbb {A}}_F)$ such that $r_\iota (\pi _1) \cong \rho _1$ and $r_\iota (\pi _2) \cong \rho _2$. Suppose moreover that $\rho _1 \otimes \rho _2$ is Hodge–Tate regular (i.e. for each embedding $\tau {:}\,F \hookrightarrow \overline{{\mathbb {Q}}}_p$, the set $\mathrm {HT}_\tau (\rho _1 \otimes \rho _2)$ contains 4 distinct elements). Then there exists a RAESDC automorphic representation $(\sigma , \psi )$ of ${{\mathrm{GL}}}_4({\mathbb {A}}_F)$ such that $r_\iota (\sigma ) \cong \rho _1 \otimes \rho _2$, where $\psi = \chi _1 \chi _2 \Vert \cdot \Vert _F$.

• (ii) Suppose that there exists a RAESDC automorphic representation $(\sigma , \psi )$ of ${{\mathrm{GL}}}_4({\mathbb {A}}_F)$ such that $r_\iota (\sigma ) \cong \rho _1 \otimes \rho _2$ and $r_\iota (\psi ) = (\det \rho _1)(\det \rho _1) \epsilon ^3$. Suppose moreover that there exist Galois totally real extensions $F_1{/}F,\,F_2{/}F$ and RAESDC automorphic representations $(\Pi _1, \xi _1)$ and $(\Pi _2, \xi _2)$ of ${{\mathrm{GL}}}_2({\mathbb {A}}_{F_1})$ and ${{\mathrm{GL}}}_2({\mathbb {A}}_{F_2})$, respectively, such that $r_\iota (\Pi _1) \cong \rho _1|_{G_{F_1}}$ and $r_\iota (\Pi _2) \cong \rho _2|_{G_{F_2}}$. Then there exists a RAESDC automorphic representation $(\pi _1, \chi _1)$ of ${{\mathrm{GL}}}_2({\mathbb {A}}_F)$ such that $r_\iota (\pi _1) \cong \rho _1$.

Proof

The first part follows from [27, Theorem M] (take $\sigma = \pi _1 \boxtimes \pi _2$). For the second part, let S be a finite set of places of F containing the infinite places, the places dividing p, and any places at which $F_1,\,F_2,\,\rho _1,\,\rho _2$, or $\sigma$ is ramified. Then we have an equality

$$\begin{aligned} L^S(\sigma \times \sigma ^\vee , s) = L^S(\sigma \times \sigma \psi ^{-1},s) = L^S(\sigma , {{\mathrm{Sym}}}^2 \otimes \psi ^{-1}, s) L^S(\sigma , \wedge ^2 \otimes \psi ^{-1}, s). \end{aligned}$$

Both sides have a meromorphic continuation to the entire complex plane. The left-hand side has a simple pole at $s = 1$, because $\sigma$ is cuspidal. On the other hand, we have

$$\begin{aligned} L^S(\sigma , \wedge ^2 \otimes \psi ^{-1}, s) = L^S( \iota [\wedge ^2 r_\iota (\sigma )] \otimes r_\iota (\psi )^{-1} \epsilon ^3, s), \end{aligned}$$

(7)

and decomposing the representation $\wedge ^2 r_\iota (\sigma )$ allows us to factorize this L-function as

$$\begin{aligned} = L^S(\iota {{\mathrm{Sym}}}^2 \rho _1 \otimes \det (\rho _2) r_\iota (\psi )^{-1} \epsilon ^3, s) L^S(\iota {{\mathrm{Sym}}}^2 \rho _2 \otimes \det (\rho _1) r_\iota (\psi )^{-1} \epsilon ^3, s). \end{aligned}$$

(8)

It follows from our assumptions, Lemma 6.4 and the existence of the symmetric square lifting for ${{\mathrm{GL}}}_2$ that the function in (8) is analytic and non-vanishing at the point $s = 1$. We deduce that the function $L^S(\sigma , {{\mathrm{Sym}}}^2 \otimes \psi ^{-1}, s)$ has a simple pole at $s = 1$. It then follows from [15, Corollary 3.2.1] (see also [1, Theorem 4.26]) that there exists a cuspidal automorphic representation $\Sigma$ of $\mathrm {GSpin}_4({\mathbb {A}}_F)$ with weak transfer equal to $\sigma$ (here we write $\mathrm {GSpin}_4$ for the split general spin group in 4 variables).

Let us now write $H = \mathrm {GSpin}_4,\,G = {{\mathrm{GL}}}_2 \times {{\mathrm{GL}}}_2$, and $Z_H, Z_G$ for the centres of these groups. There is a short exact sequence

the map $G \rightarrow {\mathbb {G}}_m$ being given by the formula $(g_1, g_2) \mapsto \det (g_1){/}\det (g_2)$. This identifies $Z_H$ with the group $\{ \lambda _1, \lambda _2 \in {\mathbb {G}}_m \times {\mathbb {G}}_m \mid \lambda _1^2 = \lambda _2^2 \} \subset Z_G$.

We have $G(F) \cap (H({\mathbb {A}}_F) Z_G({\mathbb {A}}_F)) = H(F) Z_G(F)$. The argument of [25, Proposition 3.1.4] then allows us to construct a cuspidal automorphic representation $\pi ' = \pi '_1 \otimes \pi '_2$ of $G({\mathbb {A}}_F) = {{\mathrm{GL}}}_2({\mathbb {A}}_F) \times {{\mathrm{GL}}}_2({\mathbb {A}}_F)$ with the property that $\pi '|_{H({\mathbb {A}}_F)}$ contains $\Sigma$. The representations $\pi _1', \pi '_2$ are necessarily essentially square-integrable at infinity, and it follows from [25, Proposition 2.6.7] that there exist continuous homomorphisms $R_1, R_2{:}\,G_F \rightarrow {\mathrm {PGL}}_2(\overline{{\mathbb {Q}}}_l)$ with the following properties:

• The representations $R_1, R_2$ are unramified at all finite places $v \not \in S$ of F.

• For all $v \not \in S$, we have $\mathrm {WD}(R_1|_{G_{F_v}})^{\text {F-ss}} \cong {{\mathrm{ad}}}^0 {{\mathrm{rec}}}^T_{F_v}(\iota ^{-1} \pi '_{1, v})$ and $\mathrm {WD}(R_2|_{G_{F_v}})^{\text {F-ss}} \cong {{\mathrm{ad}}}^0 {{\mathrm{rec}}}^T_{F_v}(\iota ^{-1} \pi '_{2, v})$.

Combining [25, Corollary 3.1.6] with the Chebotarev density theorem, we see that there is an isomorphism $R_1 \oplus R_2 \cong {{\mathrm{ad}}}^0 \rho _1 \oplus {{\mathrm{ad}}}^0 \rho _2$, hence (after possibly swapping $R_1$ and $R_2$) we can identify $R_1 \cong {{\mathrm{ad}}}^0 \rho _1$ and $R_2 \cong {{\mathrm{ad}}}^0 \rho _2$.

The existence of the representation $\rho _1$ implies that $\pi '_1$ has a twist $\pi ''_1$ which is regular algebraic, hence RAESDC (see [25, Proposition 2.6.7]). The representations $\rho _1$ and $r_\iota (\pi ''_1)$ satisfy ${{\mathrm{ad}}}^0 \rho _1 \cong {{\mathrm{ad}}}^0 r_\iota (\pi ''_1)$, which implies that there is a character $\eta {:}\,G_F \rightarrow \overline{{\mathbb {Q}}}_l^\times$ such that $\rho _1 \otimes \eta \cong r_\iota (\pi ''_1)$. To complete the proof of the proposition, we just need to show that the character $\eta$ is geometric (so that $\rho _1 = r_\iota (\pi ''_1) \otimes \eta ^{-1}$ is itself automorphic). However, this follows from the existence of an inclusion $\eta \subset \rho _1^\vee \otimes r_\iota (\pi _1'')$ and the fact that the property of being de Rham is stable under tensor products. $\square$

We now construct an auxiliary automorphic Galois representation which has the right local properties at 2.

Proposition 6.6

Let F be a totally real number field, and let K / F be a Galois extension. Then there exists a soluble totally real extension $F'{/}F$, an isomorphism $\iota {:}\,\overline{{\mathbb {Q}}}_2 \cong {\mathbb {C}}$, and a RAESDC automorphic representation $(\pi , \chi )$ of ${{\mathrm{GL}}}_2({\mathbb {A}}_{F'})$ satisfying the following conditions:

• (i) $\pi$ is $\iota$-ordinary and unramified at every finite place.

• (ii) $\pi$ has weight 0 and is of trivial central character.

• (iii) The residual representation $\overline{r_\iota (\pi )}$ is absolutely irreducible, with projective image conjugate to ${\mathrm {PGL}}_2({\mathbb {F}}_4)$. For each place $v | \infty$ of $F',\,\overline{r_\iota (\pi )}(c_v)$ is non-trivial.

• (iv) The extension $F'({{\mathrm{ad}}}\overline{r_\iota (\pi )}){/}F$ is linearly disjoint from K.

Proof

After possibly replacing F by a preliminary soluble extension, we can find an everywhere unramified and totally odd representation $\rho {:}\,G_F \rightarrow {{\mathrm{GL}}}_2({\mathbb {C}})$ with projective image isomorphic to $A_5$. It then follows from [26, Théorème 0.3] that $\rho$ is automorphic (i.e. associated to a cuspidal automorphic representation of ${{\mathrm{GL}}}_2({\mathbb {A}}_F)$ of limit of discrete series type at infinity). The representation $\overline{\rho }$ is absolutely irreducible, with projective image a conjugate of $A_5 \cong {\mathrm {PGL}}_2({\mathbb {F}}_4)$ (of course, $\overline{\rho }$ makes sense because of our fixed identification $\iota {:}\,\overline{{\mathbb {Q}}}_2 \cong {\mathbb {C}}$). By Hida theory, in the guise of [33, § 1], we can find an $\iota$-ordinary RAESDC automorphic representation $(\sigma , \psi )$ of weight 0 such that $\overline{r_\iota (\pi )} \cong \overline{\rho }$. Finally, passing to a further extension $F'{/}F$ and arguing as in [12, Lemma 5.1.7], we can replace the base change of $\pi$ to $F'$ by another RAESDC automorphic representation which satisfies all the requirements of the proposition. $\square$

In the next proposition, we use potential automorphy techniques as in [30] to construct automorphic ordinary lifts of our residual Galois representation of interest.

Proposition 6.7

Let k be a finite field of characteristic 2, let F be a totally real field, and let $\overline{\rho }{:}\,G_F \rightarrow {{\mathrm{GL}}}_2(k)$ be an absolutely irreducible representation with non-soluble image. Let K / F be a Galois extension. Then we can find a Galois totally real extension $F'{/}F$, an isomorphism $\iota {:}\,\overline{{\mathbb {Q}}}_2 \cong {\mathbb {C}}$, and a RAESDC automorphic representation $(\pi , \chi )$ of ${{\mathrm{GL}}}_2({\mathbb {A}}_{F'})$ satisfying the following conditions:

• (i) $\pi$ is $\iota$-ordinary and unramified at every finite place.

• (ii) $\pi$ has weight 0 and is of trivial central character.

• (iii) There is an isomorphism $\overline{r_\iota (\pi )} \cong \overline{\rho }|_{G_{F'}}$.

• (iv) The extension $F'{/}F$ is linearly disjoint from K.

Proof

Let $N{/}{\mathbb {Q}}$ be a totally real number field of degree $[k{:}\,{\mathbb {Q}}]$ in which 2 is totally inert. Let x be the unique place of N above 2, and fix an identification $k(x) \cong k$. We recall (following [30, § 1]) that a N-HBAV over a field K is a triple (A, i, j), where:

• A is an abelian variety over K of dimension $[N{:}\,{\mathbb {Q}}]$.

• $\iota$ is an embedding $\iota {:}\,{\mathcal {O}}_N \hookrightarrow {{\mathrm{End}}}_K(A)$.

• j is an isomorphism ${\mathcal {O}}_N^+ \cong {\mathcal {P}}(A, i)$ of ordered invertible ${\mathcal {O}}_N$-modules.

(an ordered invertible ${\mathcal {O}}_N$-module is an invertible ${\mathcal {O}}_N$-module M with choice of connected component of $M \otimes _{\mathbb {Q}}{\mathbb {R}}$; ${\mathcal {P}}(A, i)$ is the module of ${\mathcal {O}}_N$-linear polarizations of A; ${\mathcal {O}}_N^+$ is the standard invertible ${\mathcal {O}}_N$-module given by the identity component of ${\mathcal {O}}_N \otimes _{\mathbb {Q}}{\mathbb {R}}$).

Let $p > 5$ be a prime split in N and unramified in F, and let E / F be an elliptic curve such that the mod p Galois representation $\overline{\rho }_{E, p}$ of E has image ${{\mathrm{GL}}}_2({\mathbb {F}}_p)$, the mod 3 Galois representation $\overline{\rho }_{E, 3}$ has image ${{\mathrm{GL}}}_2({\mathbb {F}}_3)$, and E has ordinary reduction at each place v|3 of F. Let y be a place of N above p, so that $k(y) = {\mathbb {F}}_p$. Let $V_x$ be the F-scheme in k(x)-vector spaces corresponding to $\overline{\rho }$, and fix a choice of non-degenerate, Galois equivariant, symplectic pairing $V_x \times V_x \rightarrow k(x)(1)$. Let $V_y$ be the F-scheme in k(y)-vector spaces corresponding to E[p], and fix a choice of pairing $V_y \times V_y \rightarrow k(y)(1)$ (for example, the Weil pairing of E).

There is a fine moduli space X / F for tuples $(A, i, j, \eta _x, \eta _y)$, where (A, i, j) is an N-HBAV, and $\eta _x{:}\,V_x \rightarrow A[x]$ and $\eta _y{:}\,V_y \rightarrow A[y]$ are isomorphisms which take our fixed pairings to the j(1)-Weil pairings on A[x] and A[y], respectively; see [28, § 1]. Using the complex uniformization, one sees that X is smooth and $X({\mathbb {C}})$ is connected, hence X is geometrically connected.

We now claim the following:

• For each place v|2 of F, there exists a Galois extension $L_v{/}F_v$ and a Galois-invariant subset $\Omega _v \subset X(L_v)$ such that for each $P \in \Omega _v$, the corresponding N-HBAV A over $L_v$ has good ordinary reduction.

• For each place $v|\infty$ of $F,\,X(F_v)$ is non-empty.

We establish each point in turn. We first observe that if $E'{/}F$ is any elliptic curve, then $E' \otimes _{\mathbb {Z}}{\mathcal {O}}_N$ has a canonical structure of N-HBAV. In particular, there exists an N-HBAV (A, i, j) over F which has good ordinary reduction at each place v|2 of F. Let $L_v{/}F_v$ be a Galois extension such that $A[2p],\,\overline{\rho }$, and E[p] are all trivial as $G_{F_v}$-modules; then A defines a point P of $X(L_v)$. Having good ordinary reduction is an open condition, so any sufficiently small open neighbourhood $\Omega _v \subset X(L_v)$ of P satisfies the first point above.

For the second point, it is enough to show that there exist N-HBAV $(A, i, j){/}F_v$ such that $c_v$ acts trivially on A[x] (if $\overline{\rho }(c_v)$ is trivial) or that $c_v$ acts non-trivially on A[x] (if $\overline{\rho }(c_v)$ is non-trivial). Indeed, if $v|\infty$ then there are exactly 2 isomorphism classes of representations $G_{F_v} \rightarrow {{\mathrm{GL}}}_2(k)$ and exactly 1 isomorphism class of representations $G_{F_v} \rightarrow {{\mathrm{GL}}}_2(k(y))$ with odd determinant. However, this follows from the above observation and the existence of elliptic curves $E'{/}F$ with $E'[2]$ either trivial or non-trivial as $G_{F_v}$-module.

By the theorem of Moret–Bailly (see e.g. [5, Proposition 3.1.1]), we can find a Galois totally real extension $F'{/}F$ and a point $P \in X(F')$ such that for each place v|2 of F and each place w|v of $F'$, we have $F'_w \cong L_v$ as $F_v$-algebras and $P \in \Omega _v$. In particular, writing (A, i, j) for the N-HBAV over $F'$ corresponding to P, we see that A has good ordinary reduction at each place w|2 of $F',\,A[x] \cong \overline{\rho }|_{G_{F'}}$, and $A[y] \cong \overline{\rho }_{E, p}|_{G_{F'}}$. We can moreover choose $F'{/}F$ be disjoint from K / F. By first enlarging K / F to contain the extension of F cut out by $\overline{\rho }$ and E[3p], we can therefore assume that $\overline{\rho }(G_{F'}) = G_F,\,\overline{\rho }_{E, p}(G_{F'}) = \overline{\rho }_{E, p}(G_F)$, and $\overline{\rho }_{E, 3}(G_{F'}) = \overline{\rho }_{E, 3}(G_F)$.

It follows from the theorem of Langlands–Tunnell that $\overline{\rho }_{E, 3}|_{G_{F'}}$ is modular, hence $E{/}F'$ is modular by [11, Theorem 1.1], hence $A{/}F'$ is modular by [5, Theorem 4.2.1], i.e. there exists a RAESDC automorphic representation $(\pi , \chi )$ of ${{\mathrm{GL}}}_2({\mathbb {A}}_{F'})$ and an isomorphism $\iota {:}\,\overline{{\mathbb {Q}}}_2 \cong {\mathbb {C}}$ such that $r_\iota (\pi ) \cong T_x A$. We observe that $\pi$ has weight 0, is $\iota$-ordinary and unramified at the places above 2, and has trivial central character. The proposition now follows on replacing $F'$ by a soluble totally real extension and applying the main theorem of [29]. $\square$

Lemma 6.8

Let F be a totally real number field, and let $\overline{\rho }{:}\,G_F \rightarrow {{\mathrm{SL}}}_2(\overline{{\mathbb {F}}}_2)$ be a continuous, absolutely irreducible representation with insoluble image. Suppose that $\overline{\rho }$ is everywhere unramified, and that for each place v|2 of $F,\,\overline{\rho }|_{G_{F_v}}$ is trivial. Then there exists a continuous lifting $\rho {:}\,G_F \rightarrow {{\mathrm{GL}}}_2(\overline{{\mathbb {Q}}}_2)$ of $\overline{\rho }$ satisfying the following conditions:

• (i) For each finite place $v\not \mid 2$ of $F,\,\rho |_{G_{F_v}}$ is unramified.

• (ii) For each place v|2 of F and for each embedding $\tau {:}\,F_v \hookrightarrow \overline{{\mathbb {Q}}}_2,\,\rho |_{G_{F_v}}$ is crystalline ordinary and $\mathrm {HT}_\tau (\rho ) = \{ 0, 1 \}$.

• (iii) $\det \rho = \epsilon ^{-1}$.

• (iv) There is a totally real Galois extension $F'{/}F$, a RAESDC automorphic representation $(\pi , \chi )$ of ${{\mathrm{GL}}}_2({\mathbb {A}}_{F'})$ and an isomorphism $\iota {:}\,\overline{{\mathbb {Q}}}_2 \cong {\mathbb {C}}$ such that $\rho |_{G_{F'}} \cong r_\iota (\pi )$.

Proof

This follows from Proposition 6.7 and the method of Khare–Wintenberger, as we now sketch. Fix a coefficient field E with ring of integers ${\mathcal {O}}$ and residue field k, such that $\overline{\rho }$ takes values in ${{\mathrm{GL}}}_2(k)$. Let us say that a totally real extension $F'{/}F$ is allowable if $F'$ and $F(\overline{\rho })$ are linearly disjoint over F. If $F'{/}F$ is allowable, we write ${\mathcal {D}}^\square _{F'}{:}\,{\mathrm {CNL}}\rightarrow \mathrm {Sets}$ for the functor of liftings $\rho {:}\,G_{F'} \rightarrow {{\mathrm{GL}}}_2(R)$ of $\overline{\rho }|_{G_{F'}}$ satisfying the following conditions:

• If $v \not \mid 2$ is a place of $F'$, then $\rho |_{G_{F'_v}}$ is unramified.

• If v | 2 is a place of $F'$, then $\rho |_{G_{F'_v}}$ is crystalline and ordinary with Hodge–Tate weights $\{0, 1\}$. More precisely, $\rho |_{G_{F'_v}}^\vee$ defines a point of the ring $R^{\text {ord}, 1, \square }_{\overline{\rho }}$ which is defined in [18, Proposition 2.4.6]. We note that it is shown in loc. cit. that under our hypotheses, the ring $R^{\text {ord}, 1, \square }_{\overline{\rho }}$ is a domain.

• $\det \rho = \epsilon ^{-1}$.

We write ${\mathcal {D}}_{F'}$ for the corresponding functor of deformations (i.e. quotient of ${\mathcal {D}}_{F'}^\square$ by the free action of $\widehat{{\mathrm {PGL}}}_2$). We write $R_{F'} \in {\mathrm {CNL}}_{\mathcal {O}}$ for the representing object of ${\mathcal {D}}_{F'}$ (which exists). A standard calculation in obstruction theory (cf. [21, Proposition 4.5]) shows that $\dim R_{F'} \ge 1$ for any choice of $F'$ as above.

By Proposition 6.7, we can find an allowable extension $F'{/}F$ such that $R_{F'}$ has an automorphic $\overline{{\mathbb {Q}}}_2$-point. It then follows from a patching argument (cf. [21, Proposition 9.3]) that $R_{F'}$ is a finite ${\mathcal {O}}$-algebra, and every $\overline{{\mathbb {Q}}}_2$-point of $R_{F'}$ corresponds to an automorphic Galois representation. The map $R_{F'} \rightarrow R_{F}$ is finite (cf. [19, Lemma 3.6]), so $R_F$ is a finite ${\mathcal {O}}$-algebra, so has a $\overline{{\mathbb {Q}}}_2$-point (as $\dim R_F \ge 1$) corresponding to a representation $\rho {:}\,G_F \rightarrow {{\mathrm{GL}}}_2(\overline{{\mathbb {Q}}}_2)$. We see that $\rho |_{G_{F'}}$ is automorphic. This completes the proof of the lemma. $\square$

6.1 The proof of Theorem 6.2

We can now finish the proof of Theorem 6.2. Recall that in the situation of the theorem, F is a totally real number field and $\overline{\rho }_1{:}\,G_F \rightarrow {{\mathrm{GL}}}_2(\overline{{\mathbb {F}}}_2)$ is a continuous, absolutely irreducible representation with insoluble image. Moreover, there is an isomorphism $\iota {:}\,\overline{{\mathbb {Q}}}_2 \cong {\mathbb {C}}$ and a RAESDC automorphic representation $(\pi _1, \chi _1)$ such that $\overline{r_\iota (\pi _1)} \cong \overline{\rho }_1$.

After replacing F by a soluble totally real extension and $\pi _1$ by another automorphic representation, we can assume the following:

• The representation $\pi _1$ has weight 0 and trivial central character. For each finite place v of $F,\,\pi _{1, v}$ is unramified. For each place v|2 of $F,\,\pi _{1, v}$ is not $\iota$-ordinary and $\overline{\rho }_1|_{G_{F_v}}$ is trivial. Let $\rho _1 = r_\iota (\pi _1)$.

• There exists a continuous lift $\rho _1'{:}\,G_F \rightarrow {{\mathrm{GL}}}_2(\overline{{\mathbb {Q}}}_2)$ of $\overline{\rho }_1$, unramified outside 2, crystalline-ordinary above 2, and with $\mathrm {HT}_\tau (\rho _1') = \{ 0, 2 \}$ for each embedding $\tau {:}\,F \hookrightarrow \overline{{\mathbb {Q}}}_2$. Moreover, $\rho _1'$ is potentially automorphic (use Lemma 6.8, and Hida theory to change the weight).

We can also assume given the following:

• A continuous, everywhere unramified representation $\overline{\rho }_2{:}\,G_F \rightarrow {{\mathrm{GL}}}_2(\overline{{\mathbb {F}}}_2)$, with projective image conjugate to ${\mathrm {PGL}}_2({\mathbb {F}}_4)$. For each place $v | 2,\,\overline{\rho }_2|_{G_{F_v}}$ is trivial and for each place $v|\infty ,\,\overline{\rho }_2(c_v)$ is non-trivial. Moreover, the extensions $F({{\mathrm{ad}}}\overline{\rho }_1)$ and $F({{\mathrm{ad}}}\overline{\rho }_2)$ are linearly disjoint over F (use Proposition 6.6).

• A RAESDC automorphic representation $(\pi _2, \chi _2)$ of ${{\mathrm{GL}}}_2({\mathbb {A}}_F)$ which is everywhere unramified, of trivial central character, $\iota$-ordinary, and of weight ${\varvec{\lambda }}= (\lambda _\tau )_{\tau \in {{\mathrm{Hom}}}(F, {\mathbb {C}})},\,\lambda _\tau = (1, 0)$ for all $\tau$. Moreover, we have $\overline{r_\iota (\pi _2)} \cong \overline{\rho }_2$. We write $\rho _2 = r_\iota (\pi _2)$ (use Proposition 6.6, and Hida theory to change the weight).

• A RAESDC automorphic representation $(\pi _2', \chi _2')$ of ${{\mathrm{GL}}}_2({\mathbb {A}}_F)$ which is everywhere unramified, of trivial central character, and of weight 0. Moreover, we have $\overline{r_\iota (\pi '_2)} \cong \overline{\rho }_2$ and for each place v|2 of $F,\,\pi '_{2, v}$ is not $\iota$-ordinary. We write $\rho _2' = r_\iota (\pi '_2)$ (the existence of $\pi '_2$ is can be deduced from that of $\pi _2$, together with our freedom to make a base change).

Let $r = \rho _1 \otimes \rho _2'$, and $r' = \rho _1' \otimes \rho _2$. Then we have $\overline{r} \cong \overline{r}' \cong \overline{\rho }_1 \otimes \overline{\rho }_2$. To prove the theorem, it is enough to show that the representation $r'$ is automorphic. Indeed, it then follows from the second part of Proposition 6.5 that $\rho _1'$ is automorphic, and another application of Hida theory implies the existence of an ordinary weight 0 automorphic lift of $\overline{\rho }_1$.

On the other hand, we know by the first part of Proposition 6.5 that the representation r is automorphic. We will now apply Theorem 5.1 to deduce the automorphy of $r'$ from that of r. Let E / F be a totally imaginary quadratic extension in which every place v|2 of F splits. Let $S_2$ denote the set of places of F above 2, and choose for each place $v \in S_2$ a place ${\widetilde{v}}$ of E above v. Let $\widetilde{S}_2 = \{ {\widetilde{v}}\mid v \in S_2 \}$. By [5, Lemma A.2.5], we can find a crystalline character $\mu {:}\,G_E \rightarrow \overline{{\mathbb {Q}}}_2^\times$ such that $\mu \mu ^c = \epsilon ^{-1}$ and for each embedding $\tau {:}\,E \hookrightarrow \overline{{\mathbb {Q}}}_2$, we have $\mathrm {HT}_\tau (\mu ) = 0$ if $\tau$ induces a place of $\widetilde{S}_2$, and $\mathrm {HT}_\tau (\mu ) = 1$ otherwise. We now claim the following:

• There exists a RACSDC automorphic representation $\sigma$ of ${{\mathrm{GL}}}_4({\mathbb {A}}_E)$ such that $r_\iota (\sigma ) \cong r|_{G_E} \otimes \mu$. Indeed, we can take $\sigma = (\pi _1 \boxtimes \pi _2')_E \otimes (|\cdot |^{1{/}2}\iota ^{-1}\mu \circ \det )$, where $(?)_E$ denotes quadratic base change to E.

• The group $\overline{r}(G_E) \subset {{\mathrm{GL}}}_4(\overline{{\mathbb {F}}}_2)$ is adequate. This is the content of Lemma 6.3.

• If v is a place of E, then $r|_{G_E} \otimes \mu \sim r'|_{G_E} \otimes \mu$ (see Sect. 3.2 for the definition of $\sim$). If $v \not \mid 2$, then this is clear (as $r|_{G_{E_v}}$ and $r'|_{G_{E_v}}$ are unramified, so $r|_{G_{E_v}} \sim r'|_{G_{E_v}}$, and this relation is preserved under character twists, cf. the remarks after [5, Lemma 1.3.3]). If v | 2, then we observe that $\rho _1|_{G_{F_v}} \sim \rho _2|_{G_{F_v}}$ (by [18, Corollary 2.3.13]) and $\rho _1'|_{G_{F_v}} \sim \rho '_2|_{G_{F_v}}$ (by [12, Lemma 3.4.3]). The remarks in [5, § 1.4] then imply that $(\rho _1 \otimes \rho _2')|_{G_{F_v}} \sim (\rho _1' \otimes \rho _2)|_{G_{F_v}}$, hence $(r \otimes \mu )|_{G_{E_{\widetilde{v}}}} \sim (r' \otimes \mu )|_{G_{E_{\widetilde{v}}}}$.

• Let $v|\infty$ be a place such that $\overline{\rho }_1(c_v)$ is non-trivial (which exists, by hypothesis). The representation $(\overline{r}, \epsilon ^{-3})$ is polarized, and is strongly residually odd at v (the proof is the same as the proof of Lemma 3.5, given that $\overline{r}(c_v)$ is the tensor product of two regular unipotent matrices).

The hypotheses of Theorem 5.1 are therefore satisfied, and we deduce that the representation $r'|_{G_E} \otimes \mu$ is automorphic, hence $r' = \rho _1' \otimes \rho _2$ is automorphic, by soluble descent. As indicated above, this implies the desired result.

7 An erratum to [32]

In this section we correct a mistake in [32]. There is a gap in the reduction of [32, Theorem 7.1] to [32, Theorem 6.8]. The same gap appears in the reduction of [32, Theorem 9.1] to [32, Theorem 8.6]. The problem arises as follows. Let l be a prime, let F be an imaginary CM number field with maximal totally real subfield $F^+$, and let $r{:}\,G_{F^+} \rightarrow {\mathcal {G}}_n(\overline{{\mathbb {F}}}_l)$ be a continuous representation such that $r^{-1}({\mathcal {G}}_n^0(\overline{{\mathbb {F}}}_l)) = G_F$, and $\overline{\rho } = \overline{r}|_{G_F}$ is absolutely irreducible. If $F \subset F^+(\zeta _l)$, then $G_{F^+(\zeta _l)} \subset G_F$, and it is not possible for the subgroup $\overline{r}(G_{F^+(\zeta _l)}) \subset {\mathcal {G}}_n(\overline{{\mathbb {F}}}_l)$ to be adequate in the sense of [32, Definition 2.3] (as this definition requires in particular that the subgroup surject onto the component group of ${\mathcal {G}}_n$).

The condition that $\overline{r}(G_{F^+(\zeta _l)}) \subset {\mathcal {G}}_n(\overline{{\mathbb {F}}}_l)$ be an adequate subgroup is needed in the proof of [32, Theorem 6.8] in order to be able to invoke [32, Proposition 4.4]. The hypotheses of [32, Theorem 7.1] do not rule out the possibility that $F \subset F^+(\zeta _l)$. This problem does not arise in e.g. the proof of [8, Theorem 4.4.2] since it is assumed there that the prime l is unramified in F, so in particular $F \cap {\mathbb {Q}}(\zeta _l)$ must be trivial.

In order to correct the mistake, we therefore state and prove a revised version of [32, Proposition 4.4] which holds under more general conditions. This proposition is very similar to Proposition 2.21. We use notation and deformation-theoretic definitions as in [32]. In particular, l is a fixed odd prime.

Proposition 7.1

Let $q_0 \in {\mathbb {Z}}_{\ge 0}$ and suppose that $\zeta _l \not \in F$. Suppose given a deformation problem

$$\begin{aligned} {\mathcal {S}}= (F{/}F^+, S, \widetilde{S}, {\mathcal {O}}, \overline{r}, \chi , \{ {\mathcal {D}}_v \}_{v \in S}). \end{aligned}$$

Suppose that $\overline{\rho } = \overline{r}|_{G_F}$ is absolutely irreducible and that $\overline{\rho }(G_{F(\zeta _l)}) \subset {{\mathrm{GL}}}_n(\overline{{\mathbb {F}}}_l)$ is adequate, in the sense of Definition 2.20. Suppose also that for $v \in S - T$ we have

$$\begin{aligned} \dim _k L_v - \dim _k H^0(G_{F_{\widetilde{v}}}, {{\mathrm{ad}}}\overline{r}) = \left\{ \begin{array}{ll} [F_v^+{:}\,{\mathbb {Q}}_l] n (n-1){/}2 &{}\quad \text {if }v | l,\\ 0 &{}\quad \text {if }v | l. \end{array}\right. \end{aligned}$$

Let q be the larger of $\dim _k H^1_{{\mathcal {L}}^\perp , T}(G_{F^+, S}, {{\mathrm{ad}}}\overline{r}(1))$ and $q_0$.

Then for any $N \in {\mathbb {Z}}_{\ge 1}$, we can find $(Q, \widetilde{Q}, \{ \overline{\psi }_v \}_{v \in Q} )$ as in [32, Definition 4.1], such that

• $\# Q = q \ge q_0$.

• if $v \in Q$ then ${\mathbb {N}}v \equiv 1 \text { mod }l^N$.

• $R_{{\mathcal {S}}_Q}^{\square _T}$ can be topologically generated over $R_{{\mathcal {S}}, T}^\text {loc} = R_{{\mathcal {S}}_Q, T}^\text {loc}$ by

  $$\begin{aligned} \# Q - \sum _{v \in T, v| l} [F_v^+{:}\,{\mathbb {Q}}_l] n (n-1){/}2 - n \sum _{v | \infty } (1 + \chi (c_v)){/}2 \end{aligned}$$

  elements.

Proof

Fix $N \ge 1$ and let $\overline{\rho } = \overline{r}|_{G_F}$. Just as in the proof of [32, Proposition 4.4], we can reduce to showing the following claim: for any element $[\phi ] \in H^1_{{\mathcal {L}}^\perp , T}(G_{F^+, S}, {{\mathrm{ad}}}\overline{r}(1))$, we can find an element $\sigma \in G_{F(\zeta _{l^N})}$ such that $\overline{\rho }(\sigma )$ is semi-simple, together with an eigenvalue $\alpha$ of $\overline{\rho }(\sigma )$ such that ${{\mathrm{tr}}}e_{\sigma , \alpha } \phi (\sigma ) \ne 0$.

Let L be the extension of $F^+(\zeta _{l^N})$ cut out by $\overline{r}$; equivalently, the extension of $F(\zeta _{l^N})$ cut out by $\overline{\rho }$. We first show that

$$\begin{aligned} H^1({{\mathrm{Gal}}}(L{/}F^+), {{\mathrm{ad}}}\overline{r}(1)) = 0. \end{aligned}$$

(9)

There is a short exact sequence

(10)

Since $\overline{\rho }(G_{F(\zeta _l)})$ is adequate, it has no l-power order quotients. We find that $\overline{\rho }(G_{F(\zeta _{l^N})}) = \overline{\rho }(G_{F(\zeta _l)})$ and ${{\mathrm{ad}}}\overline{r}(1)^{G_{F(\zeta _{l^N})}} \cong k(\epsilon \delta _{F{/}F^+})$. The first term of (10) can therefore be identified with

$$\begin{aligned}&H^1({{\mathrm{Gal}}}(F(\zeta _{l^N}){/}F(\zeta _l)), k)^{\epsilon \delta _{F{/}F^+}} = \left\{ f \in {{\mathrm{Hom}}}({{\mathrm{Gal}}}(F(\zeta _{l^N}){/}F(\zeta _l)), k) \mid \right. \\&\quad \forall x \in \left. {{\mathrm{Gal}}}(F(\zeta _{l^N}){/}F(\zeta _l)), y \in {{\mathrm{Gal}}}(F(\zeta _{l^N}){/}F^+), f(y x y^{-1}) = \epsilon \delta _{F{/}F^+}(y) f(x) \right\} . \end{aligned}$$

Since the extension $F(\zeta _{l^N}){/}F^+$ is abelian, this group can be non-zero if and only if the character $\epsilon \delta _{F{/}F^+}$ is trivial on $G_{F^+}$, if and only if $F = F^+(\zeta _l)$. However, we have assumed that $\zeta _l \not \in F$, so we find that this group is zero. The third term of (10) is also zero, because ${{\mathrm{Gal}}}(L{/}F(\zeta _{l^N})) \cong \overline{\rho }(G_{F(\zeta _l)})$ and this last subgroup is assumed to be adequate.

There is another short exact sequence

(11)

We have shown that the first term of (11) is 0, and hence the image of the cohomology class $[\phi ]$ in $H^1(G_{L}, {{\mathrm{ad}}}\overline{r}(1))^{G_{F^+}}$ is non-zero. There is an inclusion

$$\begin{aligned} H^1(G_{L}, {{\mathrm{ad}}}\overline{r}(1))^{G_{F^+}} \subset H^1(G_L, {{\mathrm{ad}}}\overline{\rho })^{G_{F(\zeta _l)}}, \end{aligned}$$

so we can identify this restriction with a non-zero, $G_{F(\zeta _l)}$-equivariant homomorphism $f{:}\,G_L \rightarrow {{\mathrm{ad}}}\overline{\rho }$.

By assumption, we can find an element $\sigma _0 \in G_{F(\zeta _{l^N})}$ such that $\overline{\rho }(\sigma _0)$ is semi-simple, together with an eigenvalue $\alpha \in k$ of $\overline{\rho }(\sigma _0)$ such that ${{\mathrm{tr}}}e_{\sigma _0, \alpha } f(G_L) \ne 0$. If ${{\mathrm{tr}}}e_{\sigma _0, \alpha } \phi (\sigma _0) \ne 0$, then we’re done on taking $\sigma = \sigma _0$. Suppose instead that ${{\mathrm{tr}}}e_{\sigma _0, \alpha } \phi (\sigma _0) = 0$, and choose $\tau \in G_L$ such that ${{\mathrm{tr}}}e_{\sigma _0, \alpha } \phi (\tau ) \ne 0$. We then take $\sigma = \tau \sigma _0$ so that $\overline{\rho }(\sigma ) = \overline{\rho }(\sigma _0)$, and calculate with the cocycle relation that

$$\begin{aligned} {{\mathrm{tr}}}e_{\sigma , \alpha } \phi (\sigma ) = {{\mathrm{tr}}}e_{\sigma _0, \alpha } \phi (\tau ) + {{\mathrm{tr}}}e_{\sigma _0, \alpha } \phi (\sigma _0) = {{\mathrm{tr}}}e_{\sigma _0, \alpha } \phi (\tau ) \ne 0. \end{aligned}$$

This concludes the proof. $\square$

We then deduce:

Proposition 7.2

The results [32, Theorem 6.8] and [32, Theorem 8.6] hold with the assumption ‘$\overline{r}(G_{L^+(\zeta _l)})$ is adequate, in the sense of [32, Definition 2.3]’ replaced with the following two assumptions:

• (i) $\zeta _l \not \in L$.

• (ii) Let $\overline{\rho } = \overline{r}|_{G_L}$. Then $\overline{\rho }(G_{L(\zeta _l)}) \subset {{\mathrm{GL}}}_n(k)$ is adequate, in the sense of Definition 2.20.

Proof

As indicated above, the assumption ‘$\overline{r}(G_{L^+(\zeta _l)})$ is adequate, in the sense of [32, Definition 2.2]’ is used only to invoke [32, Proposition 4.4]. To prove the more general result, it is therefore enough to replace references to [32, Proposition 4.4] in the proofs with references to Proposition 7.1 above. $\square$

Corollary 7.3

The results [32, Theorem 7.1] and [32, Theorem 9.1] hold with the assumption ‘$\overline{\rho }(G_{F(\zeta _l)}) \subset {{\mathrm{GL}}}_n(k)$ is adequate, in the sense of [32, Definition 2.3]’ replaced with the following assumption:

• (i) The group $\overline{\rho }(G_{F(\zeta _l)}) \subset {{\mathrm{GL}}}_n(k)$ is adequate, in the sense of Definition 2.20.

Proof

The same proofs apply verbatim, except that the results [32, Theorem 6.8] and [32, Theorem 8.6] should be replaced by the modified versions in Proposition 7.2. $\square$

We observe that if $l \not \mid n$, then [32, Definition 2.3] is equivalent to Definition 2.20. However, if l | n then a subgroup of ${{\mathrm{GL}}}_n(\overline{{\mathbb {F}}}_l)$ is never adequate in the sense of [32, Definition 2.3] (because $H^0({{\mathrm{GL}}}_n(\overline{{\mathbb {F}}}_l), {{\mathrm{ad}}}^0) \ne 0$), but there are many subgroups of ${{\mathrm{GL}}}_n(\overline{{\mathbb {F}}}_l)$ which are adequate in the sense of Definition 2.20, as follows from [13, Theorem 11.5].

It follows that in the case $l \not \mid n$, we have merely fixed a gap in the proofs of [32, Theorem 7.1] and [32, Theorem 9.1]. However, in the case l | n, Corollary 7.3 is a new result, which relies upon the new definition of adequacy given in [13].