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Automorphy lifting for residually reducible $l$-adic Galois representations, II

Published online by Cambridge University Press:  15 December 2020

Patrick B. Allen
Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec H3A 0B9, Canada
James Newton
Department of Mathematics, King's College London, Strand, London WC2R 2LS, UK
Jack A. Thorne
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK
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We revisit the paper [Automorphy lifting for residually reducible $l$-adic Galois representations, J. Amer. Math. Soc. 28 (2015), 785–870] by the third author. We prove new automorphy lifting theorems for residually reducible Galois representations of unitary type in which the residual representation is permitted to have an arbitrary number of irreducible constituents.

MSC classification

Research Article
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© The Author(s) 2020

1. Introduction

In this paper, we prove new automorphy lifting theorems for Galois representations of unitary type. Thus, we are considering representations $\rho : G_F \to \operatorname {GL}_n(\bar {{{\mathbb {Q}}}}_l)$, where $G_F$ is the absolute Galois group of a CM field $F$ and $\rho$ is conjugate self-dual, i.e. there is an isomorphism $\rho ^{c} \cong \rho ^{\vee } \otimes \epsilon ^{1-n},$ where $c \in \operatorname {Aut}(F)$ is complex conjugation. We say in this paper that such a representation is automorphic if there exists a regular algebraic, conjugate self-dual, cuspidal (RACSDC) automorphic representation $\pi$ which is matched with $\rho$ under the Langlands correspondence. (See § 1.1 below for a more precise formulation.)

We revisit the context of the paper [Reference ThorneTho15], proving theorems valid in the case that $\bar {\rho }$ is absolutely reducible, but still satisfies a certain non-degeneracy condition (we say that $\bar {\rho }$ is ‘Schur’). The first theorems of this type were proved in the paper [Reference ThorneTho15], under the assumption that $\bar {\rho }$ has only two irreducible constituents. Our main motivation here is to remove this restriction. Our results are applied to the problem of symmetric power functoriality in [Reference Newton and ThorneNT19], where they are combined with level-raising theorems to establish automorphy of symmetric powers for certain level $1$ Hecke eigenforms congruent to a theta series.

We are also able to weaken some other hypotheses in [Reference ThorneTho15], leading to the following result, which is the main theorem of this paper.

Theorem 1.1 (Theorem 6.1) Let $F$ be an imaginary CM number field with maximal totally real subfield $F^{+}$ and let $n \geq 2$ be an integer. Let $l$ be a prime and suppose that $\rho : G_F \rightarrow \mathrm {GL}_n(\bar {{{\mathbb {Q}}}}_l)$ is a continuous semisimple representation satisfying the following hypotheses.

  1. (i) $\rho ^{c} \cong \rho ^{\vee } \epsilon ^{1-n}$.

  2. (ii) $\rho$ is ramified at only finitely many places.

  3. (iii) $\rho$ is ordinary of weight $\lambda$ for some $\lambda \in ({{\mathbb {Z}}}_+^{n})^{\operatorname {Hom}(F, {{\bar {{{\mathbb {Q}}}}_l}})}$.

  4. (iv) There is an isomorphism $\bar {\rho }^{\text {ss}} \cong \bar {\rho }_1 \oplus \cdots \oplus \bar {\rho }_{d}$, where each $\bar {\rho }_i$ is absolutely irreducible and satisfies $\bar {\rho }_i^{c} \cong \bar {\rho }_i^{\vee } \epsilon ^{1-n}$, and $\bar {\rho }_i \not \cong \bar {\rho }_j$ if $i \neq j$.

  5. (v) There exists a finite place ${{\widetilde {v}}}_0$ of $F$, prime to $l$, such that $\rho |_{G_{F_{{{\widetilde {v}}}_0}}}^{\text {ss}} \cong \oplus _{i=1}^{n} \psi \epsilon ^{n-i}$ for some unramified character $\psi : G_{F_{{{\widetilde {v}}}_0}} \rightarrow \bar {{{\mathbb {Q}}}}_l^{\times }$.

  6. (vi) There exist a RACSDC representation $\pi$ of $\mathrm {GL}_n({{\mathbb {A}}}_F)$ and $\iota : {{\bar {{{\mathbb {Q}}}}_l}} \to {{\mathbb {C}}}$ such that:

    1. (a) $\pi$ is $\iota$-ordinary;

    2. (b) $\overline {r_{ \iota }(\pi )}^{\text {ss}} \cong \bar {\rho }^{\text {ss}}$;

    3. (c) $\pi _{{{\widetilde {v}}}_0}$ is an unramified twist of the Steinberg representation.

  7. (vii) $F(\zeta _l)$ is not contained in $\bar {F}^{\ker \rm ad (\bar {\rho }^{\text {ss}})}$ and $F$ is not contained in $F^{+}(\zeta _l)$. For each $1 \leq i, j \leq {d}$, $\bar {\rho }_i|_{G_{F(\zeta _l)}}$ is absolutely irreducible and $\bar {\rho }_i|_{G_{F(\zeta _l)}} \not \cong \bar {\rho }_j|_{G_{F(\zeta _l)}}$ if $i \neq j$. Moreover, $\bar {\rho }^{\text {ss}}$ is primitive (i.e. not induced from any proper subgroup of $G_F$) and $\bar {\rho }^{\text {ss}}(G_{F})$ has no quotient of order $l$.

  8. (viii) $l > 3$ and $l \nmid n$.

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

Comparing this with [Reference ThorneTho15, Theorem 7.1], we see that we now allow an arbitrary number of irreducible constituents, while also removing the requirement that the individual constituents are adequate (in the sense of [Reference ThorneTho12]) and potentially automorphic. This assumption of potential automorphy was used in [Reference ThorneTho15], together with the Khare–Wintenberger method, to get a handle on the quotient of the universal deformation ring of $\bar {\rho }$ corresponding to reducible deformations. This made generalizing [Reference ThorneTho15, Theorem 7.1] to the case where more than two irreducible constituents are allowed seem a formidable task: one would want to know that any given direct sum of irreducible constituents of $\bar {\rho }$ was potentially automorphic, and then perhaps use induction on the number of constituents to control the reducible locus.

The first main innovation in this paper that allows us to bypass this is the observation that by fully exploiting the ‘connectedness dimension’ argument to prove that $R = {{\mathbb {T}}}$ (which goes back to [Reference Skinner and WilesSW99] and appears in this paper in the proof of Theorem 5.1), one only needs to control the size of the reducible locus in quotients of the universal deformation ring that are known a priori to be finite over the Iwasawa algebra $\Lambda$. This can be done easily by hand using the ‘locally Steinberg’ condition (as in § 3.3).

The second main innovation is a finer study of the universal deformation ring $R^{\text {univ}}$ of a (reducible but) Schur residual representation. We show that if the residual representation has ${d}$ absolutely irreducible constituents, then there is an action of a group $\mu _2^{d}$ on $R^{\text {univ}}$ and identify the invariant subring $(R^{\text {univ}})^{\mu _2^{d}}$ with the subring topologically generated by the traces of Frobenius elements (which can also be characterized as the image $P$ of the canonical map to $R^{\text {univ}}$ from the universal pseudodeformation ring). This leads to a neat proof that the map $P \to R^{\text {univ}}$ is étale at prime ideals corresponding to irreducible deformations of $\bar {\rho }$.

We now describe the organization of this paper. Since it is naturally a continuation of [Reference ThorneTho15], we maintain the same notation and use several results and constructions from that paper as black boxes. We begin in §§ 2 and 3 by extending several results from [Reference ThorneTho15] about the relation between deformations and pseudodeformations to the case where $\bar {\rho }$ is permitted to have more than two irreducible constituents. We also make the above-mentioned study of the dimension of the locus of reducible deformations.

In § 4 we recall from [Reference ThorneTho15] the definition of the unitary group of automorphic forms and Hecke algebras that we use, and state the ${{\mathbb {T}}}_{{\mathfrak q}} = R_{{\mathfrak p}}$ type result proved in that paper (here ${{\mathfrak p}}$ denotes a dimension 1, characteristic $l$ prime of $R$ with good properties, in particular that the associated representation to $\operatorname {GL}_n(\operatorname {Frac} R / {{\mathfrak p}})$ is absolutely irreducible). In § 5 we carry out the main argument, based on the notion of connectedness dimension, which is described above. Finally, in § 6 we deduce Theorem 1.1, following a simplified version of the argument in [Reference ThorneTho15, § 7] that no longer makes reference to potential automorphy.

1.1 Notation

We use the same notation and normalizations for Galois groups, class field theory, and local Langlands correspondences as in [Reference ThorneTho15, Notation]. Rather than repeat this verbatim here we invite the reader to refer to that paper for more details. We do note the convention that if $R$ is a ring and $P$ is a prime ideal of $R$, then $R_{(P)}$ denotes the localization of $R$ at $P$ and $R_P$ denotes the completion of the localization.

We recall that ${{\mathbb {Z}}}_n^{+} \subset {{\mathbb {Z}}}^{n}$ denotes the set of tuples $\lambda = (\lambda _1, \ldots , \lambda _n)$ of integers such that $\lambda _1 \geq \cdots \geq \lambda _n$. It is identified in a standard way with the set of highest weights of $\operatorname {GL}_n$. If $F$ is a number field and $\lambda = (\lambda _\tau ) \in ({{\mathbb {Z}}}_n^{+})^{\operatorname {Hom}(F, {{\mathbb {C}}})}$, then we write $\Xi _\lambda$ for the algebraic representation of $\operatorname {GL}_n(F \otimes _{{\mathbb {Q}}} {{\mathbb {C}}}) = \prod _{\tau \in \operatorname {Hom}(F, {{\mathbb {C}}})} \operatorname {GL}_n({{\mathbb {C}}})$ of highest weight $\lambda$. If $\pi$ is an automorphic representation of $\operatorname {GL}_n({{\mathbb {A}}}_F)$, we say that $\pi$ is regular algebraic of weight $\lambda$ if $\pi _\infty$ has the same infinitesimal character as the dual $\Xi _\lambda ^{\vee }$.

Let $F$ be a CM field (i.e. a totally imaginary quadratic extension of a totally real field $F^{+}$). We always write $c \in \operatorname {Aut}(F)$ for complex conjugation. We say that an automorphic representation $\pi$ of $\operatorname {GL}_n({{\mathbb {A}}}_F)$ is conjugate self-dual if there is an isomorphism $\pi ^{c} \cong \pi ^{\vee }$. If $\pi$ is a RACSDC automorphic representation of $\operatorname {GL}_n({{\mathbb {A}}}_F)$ and $\iota : \bar {{{\mathbb {Q}}}}_l \to {{\mathbb {C}}}$ is an isomorphism (for some prime $l$), then there exists an associated Galois representation $r_\iota (\pi ) : G_F \to \operatorname {GL}_n(\bar {{{\mathbb {Q}}}}_l)$, characterized up to isomorphism by the requirement of compatibility with the local Langlands correspondence at each finite place of $F$; see [Reference ThorneTho15, Theorem 2.2] for a reference. We say that a representation $\rho : G_F \to \operatorname {GL}_n(\bar {{{\mathbb {Q}}}}_l)$ is automorphic if there exists a choice of $\iota$ and RACSDC $\pi$ such that $\rho \cong r_\iota (\pi )$.

One can define what it means for a RACSDC automorphic representation $\pi$ to be $\iota$-ordinary (see [Reference ThorneTho15, Lemma 2.3]; it means that the eigenvalues of certain Hecke operators, a priori $l$-adic integers, are in fact $l$-adic units). If $\mu \in ({{\mathbb {Z}}}^{n}_+)^{\operatorname {Hom}(F, \bar {{{\mathbb {Q}}}}_l)}$, we say (following [Reference ThorneTho15, Definition 2.5]) that a representation $\rho : G_F \to \operatorname {GL}_n(\bar {{{\mathbb {Q}}}}_l)$ is ordinary of weight $\mu$ if for each place $v | l$ of $F$, there is an isomorphism

\[ \rho|_{G_{F_v}} \sim \begin{pmatrix} \psi_1 & \ast & \ast & \ast \\ 0 & \psi_2 & \ast & \ast \\ \vdots & \ddots & \ddots & \ast \\ 0 & \ldots & 0 & \psi_n \end{pmatrix}\!, \]

where $\psi _i : G_{F_v} \rightarrow {{\bar {{{\mathbb {Q}}}}_l}}^{\times }$ is a continuous character satisfying the identity

\[ \psi_i(\sigma) = \prod_{\tau : F_v \hookrightarrow {{\bar{{{\mathbb{Q}}}}_l}}} \tau(\operatorname{Art}_{F_v}^{-1}(\sigma))^{-( \mu_{\tau, n - i + 1} + i - 1)} \]

for all $\sigma$ in a suitable open subgroup of $I_{F_v}$. An important result [Reference ThorneTho15, Theorem 2.4] is that if $\pi$ is RACSDC of weight $\lambda$ and $\iota$-ordinary, then $r_\iota (\pi )$ is ordinary of weight $\iota \lambda$, where by definition $(\iota \lambda )_\tau = \lambda _{\iota \tau }$.

2. Determinants

We first give the definition of a determinant from [Reference ChenevierChe14]. We recall that if $A$ is a ring and $M, N$ are $A$-modules, then an $A$-polynomial law $F : M \to N$ is a natural transformation $F : h_M \to h_N$, where $h_M : A$-alg $\to$ Sets is the functor $h_M(B) = M \otimes _A B$. The $A$-polynomial law $F$ is called homogeneous of degree $n \geq 1$ if for all $b \in B$, $x \in M \otimes _A B$, we have $F_B(bx) = b^{n} F_B(x)$.

Definition 2.1 Let $A$ be a ring and let $R$ be an $A$-algebra. An $A$-valued determinant of $R$ of dimension $n \geq 1$ is a multiplicative $A$-polynomial law $D : R \to A$ which is homogeneous of degree $n$.

If $D$ is a determinant, then there are associated polynomial laws $\Lambda _i : R \to A$, $i = 0, \ldots , n$, given by the formulae

\[ D(t - r) = \sum_{i=0}^{n} (-1)^{i} \Lambda_i(r) t^{n-i} \]

for all $r \in R \otimes _A B$. We define the characteristic polynomial $A$-polynomial law $\chi : R \to R$ by the formula $\chi (r) = \sum _{i=0}^{n} (-1)^{i} \Lambda _i(r) r^{n-i}$ ($r \in R \otimes _A B$). We write $\operatorname {CH}(D)$ for the two-sided ideal of $R$ generated by the coefficients of $\chi (r_1 t_1 + \cdots + r_m t_m) \in R[t_1, \ldots , t_m]$ for all $m \ge 1$ and $r_1,\ldots ,r_m \in R$. We have $\operatorname {CH}(D) \subseteq \ker (D)$ [Reference ChenevierChe14, Lemma 1.21]. The determinant $D$ is said to be Cayley–Hamilton if $\operatorname {CH}(D) = 0$, equivalently if $\chi = 0$ (i.e. $\chi$ is the zero $A$-polynomial law).

We next recall the definition of a generalized matrix algebra [Reference Bellaïche and ChenevierBC09, Definition 1.3.1].

Definition 2.2 Let $A$ be a ring and let $R$ be an $A$-algebra. We say that $R$ is a generalized matrix algebra of type $(n_1,\ldots ,n_{d})$ if it is equipped with the following data:

  1. (i) a family of orthogonal idempotents $e_1,\ldots , e_{d}$ with $e_1+\cdots +e_{d} = 1$; and

  2. (ii) for each $1\le i \le {d}$, an $A$-algebra isomorphism $\psi _i \colon e_i R e_i \rightarrow M_{n_i}(A)$

such that the trace map $T \colon R \rightarrow A$ defined by $T(x) = \sum _{i=1}^{d} \operatorname {tr}\psi _i(e_ix e_i)$ satisfies $T(xy) = T(yx)$ for all $x,y \in R$. We refer to the data $\mathcal {E} = \{e_i,\psi _i, 1\le i \le {d}\}$ as the data of idempotents of $R$.

Construction 2.3 We recall the structure of generalized matrix algebras from [Reference Bellaïche and ChenevierBC09, § 1.3.2]. Let $R$ be a generalized matrix algebra of type $(n_1,\ldots ,n_{d})$ with data of idempotents $\mathcal {E} = \{e_i,\psi _i, 1\le i \le {d}\}$. For each $1\le i \le {d}$, let $E_i \in e_i R e_i$ be the unique element such that $\psi _i(E_i)$ is the element of $M_{n_i}(A)$ whose row $1$, column $1$ entry is $1$ and all other entries are $0$. We set $\mathcal {A}_{i,j} = E_i R E_j$ for each $1\le i,j\le {d}$. Note that $\mathcal {A}_{i,j}\mathcal {A}_{j,k} \subseteq \mathcal {A}_{i,k}$ for each $1\le i,j,k\le {d}$, and the trace map $T$ induces an isomorphism $\mathcal {A}_{i,i} \cong A$ for each $1\le i \le {d}$. Via this isomorphism, we will tacitly view $\mathcal {A}_{i,j}\mathcal {A}_{j,i}$ as an ideal in $A$ for each $1\le i,j\le {d}$. With this multiplication, there is an isomorphism of $A$-algebras

(1)\begin{equation} R \cong \begin{pmatrix} M_{n_1}(A) & M_{n_1,n_2}(\mathcal{A}_{1,2}) & \cdots & M_{n_1,n_{{d}}}(\mathcal{A}_{1,{{d}}})\\ M_{n_2,n_1}(\mathcal{A}_{2,1}) & M_{n_2}(A) & \cdots & M_{n_2,n_{{d}}}(\mathcal{A}_{2,{{d}}})\\ \vdots & \vdots & \ddots & \vdots \\ M_{n_{{d}},n_1}(\mathcal{A}_{{{d}},1}) & M_{n_{{d}},n_2}(\mathcal{A}_{{{d}},2}) & \cdots & M_{n_{{d}}}(A) \end{pmatrix}\!. \end{equation}

The following result of Chenevier allows us to use the above structure when studying determinants.

Theorem 2.4 Let $A$ be a Henselian local ring with residue field $k$, let $R$ be an $A$-algebra, and let $D \colon R \rightarrow A$ be a Cayley–Hamilton determinant. Suppose that there exist surjective and pairwise non-conjugate $k$-algebra homomorphisms $\bar {\rho }_i : R \to M_{n_i}(k)$ such that $\bar {D} = \prod _{i=1}^{d} (\det \circ \bar {\rho }_i)$, where $\bar {D} = D \otimes _R k$.

Then there is a datum of idempotents $\mathcal {E} = \{e_i,\psi _i, 1\le i \le {d}\}$ for which $R$ is a generalized matrix algebra and such that $\psi _i \otimes _A k = \bar {\rho }_i|_{e_i R e_i}$. Any two such data are conjugate by an element of $R^{\times }$.

We note that the assumptions of Theorem 2.4 say that $D$ is residually split and multiplicity-free, in the sense of [Reference ChenevierChe14, Definition 2.19].

Proof. The existence of such a datum of idempotents $\mathcal {E} = \{e_i,\psi _i,1\le i \le {d}\}$ is contained in [Reference ChenevierChe14, Theorem 2.22] and its proof. The statement that two such data are conjugate is exactly as in [Reference Bellaïche and ChenevierBC09, Lemma 1.4.3]. Namely, if $\mathcal {E}' = \{e_i',\psi _i',1\le i \le {d}\}$ is another such choice, then since $\operatorname {End}_R(R e_i) \cong M_{n_i}(A) \cong \operatorname {End}_R(R e_i')$ are local rings, the Krull–Schmidt–Azumaya theorem [Reference Curtis and ReinerCR81, Theorem 6.12] (see also [Reference Curtis and ReinerCR81, Remark 6.14 and Chapter 6, Exercise 14]) implies that there is $x\in R^{\times }$ such that $x e_i x^{-1} = e_i'$ for each $1\le i \le {d}$. By Skolem–Noether, we can adjust $x$ by an element of $(\oplus _{i=1}^{d}e_i R e_i)^{\times }$ so that it further satisfies $x\psi _i x^{-1} = \psi _i'$.

We now show that the reducibility ideals of [Reference Bellaïche and ChenevierBC09, Proposition 1.5.1] and their basic properties carry over for determinants (so without having to assume that $n!$ is invertible in $A$).

Proposition 2.5 Let $A$ be a Henselian local ring with residue field $k$, let $R$ be an $A$-algebra, and let $D \colon R \rightarrow A$ be a determinant. Assume that $\bar {D} = D\otimes _A k \colon R\otimes _A k \rightarrow k$ is split and multiplicity free. Write $\bar {D} = \prod _{i=1}^{d} \bar {D}_i$ with each $\bar {D}_i$ absolutely irreducible of dimension $n_i$.

Let $\mathcal {P} = (\mathcal {P}_1,\ldots ,\mathcal {P}_s)$ be a partition of $\{1,\ldots ,{d}\}$. There is an ideal $I_\mathcal {P}$ of $A$ such that an ideal $J$ of $A$ satisfies $I_\mathcal {P} \subseteq J$ if and only if there are determinants $D_1,\ldots ,D_s \colon R\otimes _A A/J \rightarrow A/J$ such that $D\otimes _A A/J = \prod _{m=1}^{s} D_m$ and $D_m \otimes _A k = \prod _{i \in \mathcal {P}_m} \bar {D}_i$ for each $1\le m \le s$. If this property holds, then $D_1,\ldots ,D_s$ are uniquely determined and satisfy $\ker (D\otimes _A A/J) \subseteq \ker (D_m)$.

Moreover, let $\mathcal {J}$ be a two-sided ideal of $R$ with $\operatorname {CH}(D) \subseteq \mathcal {J} \subseteq \ker (D)$ and let $\mathcal {A}_{i,j}$ be the $A$-modules as in Construction 2.3 for a choice of data of idempotents as in Theorem 2.4 applied to $R/\mathcal {J}$. Then $I_\mathcal {P} = \sum _{i,j} \mathcal {A}_{i,j}\mathcal {A}_{j,i}$, where the sum is over all pairs $i,j$ not belonging to the same $\mathcal {P}_m\in \mathcal {P}$.

Proof. We follow the proof of [Reference Bellaïche and ChenevierBC09, Proposition 1.5.1] closely. Choose a two-sided ideal $\mathcal {J}$ of $R$ with $\operatorname {CH}(D) \subseteq \mathcal {J} \subseteq \ker (D)$, and data of idempotents $\mathcal {E}$ for $R/\mathcal {J}$ as in Theorem 2.4. We let $\mathcal {A}_{i,j}$ be as in Construction 2.3 and define $I_\mathcal {P} = \sum _{i,j} \mathcal {A}_{i,j}\mathcal {A}_{j,i}$, where the sum is over all pairs $i,j$ not belonging to the same $\mathcal {P}_m\in \mathcal {P}$. Since another such choice of the data of idempotents is conjugate by an element of $(R/\mathcal {J})^{\times }$, the ideal $I_{\mathcal {P}}$ does not depend on the choice of $\mathcal {E}$. To see that it is independent of $\mathcal {J}$, first note that $D$ further factors through a surjection $\psi \colon R/\mathcal {J} \rightarrow R/\ker (D)$. Under this surjection, the data of idempotents $\mathcal {E}$ is sent to a data of idempotents for $R/\ker (D)$, and $\operatorname {tr}(\psi (\mathcal {A}_{i,j})\psi (\mathcal {A}_{j,i})) = \operatorname {tr}(\mathcal {A}_{i,j}\mathcal {A}_{j,i})$ since $\operatorname {tr}\circ \psi = \operatorname {tr}$.

We can now replace $R$ with $R/\operatorname {CH}(D)$ and assume that $D$ is Cayley–Hamilton. Since $\operatorname {CH}(D)$ is stable under base change, it suffices to show that $I_{\mathcal {P}} = 0$ if and only if there are determinants $D_1,\ldots ,D_s \colon R\rightarrow A$ such that $D = \prod _{m=1}^{s} D_m$ and $D_m \otimes _A k_A = \prod _{i \in \mathcal {P}_m} \bar {D}_i$ for each $1\le m \le s$ and that, if this happens, then $D_1,\ldots ,D_s$ are uniquely determined. Fix a datum of idempotents $\mathcal {E} = \{e_i,\psi _i,1\le i \le {d}\}$ for $R$ as in Theorem 2.4, and let the notation be as in Construction 2.3. For each $1\le m \le s$, we set $f_m = \sum _{i\in \mathcal {P}_m} e_i$. Then $1 = f_1+\cdots + f_s$ is a decomposition into orthogonal idempotents.

First assume that $I_{\mathcal {P}} = 0$. Let $\tilde {D}$ denote the $A$-valued determinant on $R/\ker (D)$ arising from $D$. Fix $x \in R$, an $A$-algebra $B$, and $y \in R\otimes _A B$. If $1\le i,j\le {d}$ do not belong to the same $\mathcal {P}_m\in \mathcal {P}$, then using the algebra structure as in (1) and the fact that $\mathcal {A}_{i,j} \mathcal {A}_{j,i} = 0$, we have $e_i x e_j y = \sum _{l\ne i} e_i x e_j y e_l$, and [Reference ChenevierChe14, Lemma 1.12(i)] gives

\[ D(1 + e_i x e_ j y) = D\bigg(1+\sum_{l\ne i} e_i x e_j y e_l\bigg) = D\bigg(1+\sum_{l\ne i} x e_j y e_l e_i\bigg) = D(1) = 1. \]

By [Reference ChenevierChe14, Lemma 1.19], $e_i x e_j \in \ker (D)$ for all $x\in R$ and all $i,j$ that do not belong to the same $\mathcal {P}_m\in \mathcal {P}$. We then have an isomorphism of $A$-algebras $R/\ker (D) \cong \prod _{m=1}^{s} f_m (R/\ker (D)) f_m$ and [Reference ChenevierChe14, Lemma 2.4] gives $D = \prod _{m=1}^{s} D_m$, where $D_m \colon R \rightarrow A$ is the composite of the surjection $R \rightarrow f_m (R/\ker (D)) f_m$ with the determinant $\tilde {D}_m \colon f_m (R/\ker (D)) f_m \rightarrow A$ given by $x\mapsto \tilde {D}(x + 1-f_m)$. It is immediate that $D_m \otimes _A k_A = \prod _{i \in \mathcal {P}_m} \bar {D}_i$ for each $1\le m \le s$.

Now assume that there are determinants $D_1,\ldots ,D_s \colon R\rightarrow A$ such that $D = \prod _{m=1}^{s} D_m$ and $D_m \otimes _A k = \prod _{i \in \mathcal {P}_m} \bar {D}_i$ for each $1\le m \le s$. The determinants $D_m$ have dimension $d_m := \sum _{i\in \mathcal {P}_m} n_i$. The trace map yields an equality

\[ \sum_{1\le m\ne m' \le s} \operatorname{tr}(f_m R f_{m'}R f_m) = I_{\mathcal{P}}. \]

So, to show that $I_{\mathcal {P}} = 0$, it suffices to show that $\operatorname {tr}(f_m R f_{m'} R f_m) = 0$ for $m\ne m'$. For this, it suffices to show that $f_{m'} \in \ker (D_m)$ for any $m\ne m'$, since this implies that $f_m R f_{m'} \in \ker (D_l)$ for any $1\le l\le s$ and hence

\[ D(1+tf_m R f_{m'} R f_m) = \prod_{l=1}^{s} D_l(1+tf_m R f_{m'} R f_m) = 1. \]

For any idempotent $f$ of $R$, we have the determinant $D_{m,f} \colon f R f \rightarrow A$ given by $D_{m,f}(x) = D_m(x+1-f)$. When $f = f_m$,

\[ D_{m,f_m} \otimes_A k = \prod_{i\in \mathcal{P}_m} \bar{D}_{i,f_m} = \prod_{i\in \mathcal{P}_m}\bar{D}_{i,e_i} \]

has dimension $d_m$. Then [Reference ChenevierChe14, Lemma 2.4(2)] implies that $D_{m,1-f_m}$ has dimension $0$, i.e. is constant and equal to $1$. So, for any $m\ne m'$, the characteristic polynomial of $f_{m'}$ with respect to $D_m$ is

\[ D_m(t-f_{m'}) = D_{m,f_m}(t)D_{m,1-f_m}(t-f_{m'}) = t^{d_m}. \]

Then $f_{m'} = f_{m'}^{d_m} \in \operatorname {CH}(D_m) \subseteq \ker (D_m)$, which is what we wanted to prove. This further shows that for each $1\le m \le s$, the determinant $D_m$ is the composite of the surjections

\[ R \rightarrow \oplus_{l=1}^{s} f_l R f_l \rightarrow f_m R f_m \]

with the determinant $D_{f_m} \colon f_m R f_m \rightarrow A$. Since any two choices of the data of idempotents are conjugate under $R^{\times }$, each $D_m$ is uniquely determined by $D$.

3. Deformations

Galois deformation theory plays an essential role in this paper. The set of results we use is essentially identical to that of [Reference ThorneTho15], with some technical improvements. In this section we recall the notation used in [Reference ThorneTho15], without giving detailed definitions or proofs; we then proceed to prove the new results that we need. Some of the definitions recalled here were first given in [Reference Clozel, Harris and TaylorCHT08] or [Reference GeraghtyGer19], but in order to avoid sending the reader to too many different places we restrict our citations to [Reference ThorneTho15].

We will use exactly the same set-up and notation for deformation theory as in [Reference ThorneTho15]. We recall that this means that we fix at the outset the following objects.

  1. A CM number field $F$, with its totally real subfield $F^{+}$.

  2. An odd prime $l$ such that each $l$-adic place of $F^{+}$ splits in $F$. We write $S_l$ for the set of $l$-adic places of $F^{+}$.

  3. A finite set $S$ of finite places of $F^{+}$ which split in $F$. We assume that $S_l \subset S$ and write $F(S)$ for the maximal extension of $F$ which is unramified outside $S$ and set $G_{F, S} = \operatorname {Gal}(F(S) / F)$ and $G_{F^{+}, S} = \operatorname {Gal}(F(S) / F^{+})$. We fix a choice of complex conjugation $c \in G_{F^{+}, S}$.

  4. For each $v \in S$, we fix a choice of place ${{\widetilde {v}}}$ of $F$ such that ${{\widetilde {v}}}|_{F^{+}} = v$, and define $\tilde {S} = \{ {{\widetilde {v}}} \mid v \in S \}$.

We also fix the following data.

  1. A coefficient field $K \subset \bar {{{\mathbb {Q}}}}_l$ with ring of integers ${{\mathcal {O}}}$, residue field $k$, and maximal ideal $\lambda \subset {{\mathcal {O}}}$.

  2. A continuous homomorphism $\chi : G_{F^{+}, S} \to {{\mathcal {O}}}^{\times }$. We write $\bar {\chi } = \chi \text { mod } \lambda$.

  3. A continuous homomorphism $\bar {r} : G_{F^{+}, S} \to {{\mathcal {G}}}_n(k)$ such that $\bar {r}^{-1}({{\mathcal {G}}}_n^{\circ }(k)) = G_{F, S}$. Here ${{\mathcal {G}}}_n$ is the algebraic group over ${{\mathbb {Z}}}$ defined in [Reference Clozel, Harris and TaylorCHT08, § 2.1]. We follow the convention that if $R : \Gamma \to {{\mathcal {G}}}_n(A)$ is a homomorphism and $\Delta \subset \Gamma$ is a subgroup such that $R(\Delta ) \subset {{\mathcal {G}}}_n^{0}(A)$, then $R|_\Delta$ denotes the composite homomorphism

    \[ \Delta \to {{\mathcal{G}}}_n^{0}(A) = \operatorname{GL}_n(A) \times \operatorname{GL}_1(A) \to \operatorname{GL}_n(A). \]
    Thus, $\bar {r}|_{G_{F, S}}$ takes values in $\operatorname {GL}_n(k)$.

If $v \in S_l$, then we write $\Lambda _v = {{\mathcal {O}}} [\kern-1pt[ (I^{\text ab}_{F_{{\widetilde {v}}}}(l))^{n} ]\kern-1pt]$, where $I^{{\text ab}}_{F_{{\widetilde {v}}}}(l)$ denotes the inertia group in the maximal abelian pro-$l$ extension of $F_{{\widetilde {v}}}$. We set $\Lambda = \hat {\otimes }_v \Lambda _v$, the completed tensor product being over ${{\mathcal {O}}}$. A global deformation problem, as defined in [Reference ThorneTho15, § 3], then consists of a tuple

\[ {{\mathcal{S}}} = ( F / F^{+}, S, \tilde{S}, \Lambda, \bar{r}, \chi, \{ {{\mathcal{D}}}_v \}_{v \in S} ). \]

The extra data that we have not defined consists of the choice of a local deformation problem ${{\mathcal {D}}}_v$ for each $v \in S$. We will not need to define any new local deformation problems in this paper, but we recall that the following have been defined in [Reference ThorneTho15]:

  1. ‘ordinary deformations’ give rise to a problem ${{\mathcal {D}}}_v^{\triangle }$ for each $v \in S_l$ [Reference ThorneTho15, § 3.3.2];

  2. ‘Steinberg deformations’ give rise to a problem ${{\mathcal {D}}}_v^{\rm St}$ for each place $v \in S$ such that $q_v \equiv 1 \text { mod }l$ and $\bar {r}|_{G_{F_{{\widetilde {v}}}}}$ is trivial;

  3. $\chi _v$-ramified deformations’ give rise to a problem ${{\mathcal {D}}}_v^{\chi _v}$ for each place $v \in S$ such that $q_v \equiv 1 \text { mod }l$ and $\bar {r}|_{G_{F_{{\widetilde {v}}}}}$ is trivial, given the additional data of a tuple $\chi _v = (\chi _{v, 1}, \ldots , \chi _{v, n})$ of characters $\chi _{v, i} : k(v)^{\times }(l) \to k^{\times }$;

  4. ‘unrestricted deformations’ give rise to a problem ${{\mathcal {D}}}_v^{\square }$ for any $v \in S$.

If ${{\mathcal {S}}}$ is a global deformation problem, then we can define (as in [Reference ThorneTho15]) a functor $\operatorname {Def}_{{\mathcal {S}}} : {{\mathcal {C}}}_\Lambda \to \text {Sets}$ of ‘deformations of type ${{\mathcal {S}}}$’. By definition, if $A \in {{\mathcal {C}}}_\Lambda$, then $\operatorname {Def}_{{\mathcal {S}}}(A)$ is the set of $\operatorname {GL}_n(A)$-conjugacy classes of homomorphisms $r : G_{F^{+}, S} \to {{\mathcal {G}}}_n(A)$ lifting $\bar {r}$ such that $\nu \circ r = \chi$ and for each $v \in S$, $r|_{G_{F_{{\widetilde {v}}}}} \in {{\mathcal {D}}}_v(A)$. If $\bar {r}$ is Schur (see [Reference ThorneTho15, Definition 3.2]), then the functor $\operatorname {Def}_{{\mathcal {S}}}$ is represented by an object $R_{{\mathcal {S}}}^{\rm univ} \in {{\mathcal {C}}}_\Lambda$.

3.1 An erratum to [Reference ThorneTho15]

We point out an error in [Reference ThorneTho15]. We thank Lue Pan for bringing this to our attention. In [Reference ThorneTho15, Proposition 3.15] it is asserted that the ring $R_v^{1}$ (representing the deformation problem ${{\mathcal {D}}}_v^{1}$ for $v \in R$, defined under the assumptions $q_v \equiv 1 \text { mod }l$ and $\bar {r}|_{G_{F_{{\widetilde {v}}}}}$ trivial) has the property that $R_v^{1} / (\lambda )$ is generically reduced. This is false, even in the case $n = 2$, as can be seen from the statement of [Reference ShottonSho16, Proposition 5.8] (and noting the identification $R_v^{1} / (\lambda ) = R_v^{\chi _v} / (\lambda )$). We offer the following corrected statement.

Proposition 3.1 Let $\bar {R}_v^{1}$ denote the nilreduction of $R_v^{1}$. Then $\bar {R}_v^{1} / (\lambda )$ is generically reduced.

Proof. Let ${{\mathcal {M}}}$ denote the scheme over ${{\mathcal {O}}}$ of pairs of $n \times n$ matrices $(\Phi , \Sigma )$, where $\Phi$ is invertible, the characteristic polynomial of $\Sigma$ equals $(X - 1)^{n}$, and we have $\Phi \Sigma \Phi ^{-1} = \Sigma ^{q_v}$. Then $R_v^{1}$ can be identified with the completed local ring of ${{\mathcal {M}}}$ at the point $(1_n, 1_n) \in {{\mathcal {M}}}(k)$. By [Reference MatsumuraMat89, Theorem 23.9] (and since ${{\mathcal {M}}}$ is excellent), it is enough to show that if $\bar {{{\mathcal {M}}}}$ denotes the nilreduction of ${{\mathcal {M}}}$, then $\bar {{{\mathcal {M}}}} \otimes _{{\mathcal {O}}}k$ is generically reduced.

Let ${{\mathcal {M}}}_1, \ldots , {{\mathcal {M}}}_r$ denote the irreducible components of ${{\mathcal {M}}}$ with their reduced subscheme structure. According to [Reference ThorneTho12, Lemma 3.15], each ${{\mathcal {M}}}_i \otimes _{{\mathcal {O}}}K$ is non-empty of dimension $n^{2}$, while the ${{\mathcal {M}}}_i \otimes _{{\mathcal {O}}}k$ are the pairwise-distinct irreducible components of ${{\mathcal {M}}} \otimes _{{\mathcal {O}}}k$ and are all generically reduced. Let $\bar {\eta }_i$ denote the generic point of ${{\mathcal {M}}}_i \otimes _{{\mathcal {O}}}k$. Then $\bar {\eta }_i$ admits an open neighbourhood in ${{\mathcal {M}}}$ not meeting any ${{\mathcal {M}}}_j$ ($j \neq i$). Consequently, we have an equality of local rings ${{\mathcal {O}}}_{\bar {{{\mathcal {M}}}}, \bar {\eta }_i} = {{\mathcal {O}}}_{{{\mathcal {M}}}_i, \bar {\eta }_i}$, showing that ${{\mathcal {O}}}_{\bar {{{\mathcal {M}}}}, \bar {\eta }_i} / (\lambda )$ is reduced (in fact, a field). This shows that $\bar {{{\mathcal {M}}}} \otimes _{{\mathcal {O}}}k$ is generically reduced.

We now need to explain why this error does not affect the proofs of the two results in [Reference ThorneTho15] which rely on the assertion that $R_v^{1} / (\lambda )$ is generically reduced. The first of these is [Reference ThorneTho15, Proposition 3.17], which states that the Steinberg deformation ring $R_v^{\rm St}$ has the property that $R_v^{\rm St} / (\lambda )$ is generically reduced. The proof of this result is still valid if one replaces $R_v^{1}$ there with $\bar {R}_v^{1}$. Indeed, we need only note that $R_v^{\rm St}$ is ${{\mathcal {O}}}$-flat (by definition) and reduced (since $R_v^{\rm St}[1/l]$ is regular, by [Reference TaylorTay08, Lemma 3.3]). The map $R_v^{1} \to R_v^{\rm St}$ therefore factors through a surjection $\bar {R}_v^{1} \to R_v^{\rm St}$.

The next result is [Reference ThorneTho15, Lemma 3.40(2)], which describes the irreducible components of the localization and completion of a ring $R^{\infty }$ at a prime ideal $P_\infty$. The ring $R^{\infty }$ has $R_v^{1}$ as a (completed) tensor factor, and the generic reducedness is used to justify an appeal to [Reference ThorneTho15, Proposition 1.6]. Since passing to nilreduction does not change the underlying topological space, one can argue instead with the quotient of $R^{\infty }$, where $R_v^{1}$ is replaced by $\bar {R}_v^{1}$. The statement of [Reference ThorneTho15, Lemma 3.40] is therefore still valid.

3.2 Pseudodeformations

In this section, we fix a global deformation problem

\[ {{\mathcal{S}}} = ( F / F^{+}, S, \tilde{S}, \Lambda, \bar{r}, \chi, \{ {{\mathcal{D}}}_v \}_{v \in S} ) \]

such that $\bar {r}$ is Schur. We write $P_{{\mathcal {S}}} \subset R_{{\mathcal {S}}}^{\text {univ}}$ for the $\Lambda$-subalgebra topologically generated by the coefficients of characteristic polynomials of Frobenius elements $\operatorname {Frob}_w \in G_{F, S}$ ($w$ prime to $S$). The subring $P_{{\mathcal {S}}}$ is studied in [Reference ThorneTho15, § 3.4], where it is shown using results of Chenevier that $P_{{\mathcal {S}}}$ is a complete Noetherian local $\Lambda$-algebra and that the inclusion $P_{{\mathcal {S}}} \subset R_{{{\mathcal {S}}}}^{\rm univ}$ is a finite ring map (see [Reference ThorneTho15, Proposition 3.29]).

In fact, more is true, as we now describe. Let $\bar {B} \in \operatorname {GL}_n(k)$ be the matrix defined by the formula $\bar {r}(c) = (\bar {B}, - \chi (c))\jmath \in {{\mathcal {G}}}_n(k)$. Let $\bar {\rho } = \bar {r}|_{G_{F, S}}$ and suppose that there is a decomposition $\bar {r}=\oplus _{i=1}^{{d}} \bar {r}_i$ with $\bar {\rho }_i=\bar {r}_i|_{G_{F,S}}$ absolutely irreducible for each $i$. The representations $\bar {\rho }_i$ are pairwise non-isomorphic, because $\bar {r}$ is Schur (see [Reference ThorneTho15, Lemma 3.3]). We recall [Reference ThorneTho15, Lemma 3.1] that to give a lifting $r:G_{F^{+},S}\rightarrow {{\mathcal {G}}}_n(R)$ of $\bar {r}$ with $\nu \circ r = \chi$ is equivalent to giving the following data.

  1. A representation $\rho : G_{F,S}\rightarrow \operatorname {GL}_n(R)$ lifting $\bar {\rho } = \bar {r}|_{G_{F,S}}$.

  2. A matrix $B \in \operatorname {GL}_n(R)$ lifting $\bar {B}$ with $^{t}{}{B} = -\chi (c)B$ and $\chi (\delta )B = \rho (\delta ^{c})B ^{t}{}{\rho }(\delta )$ for all $\delta \in G_{F,S}$.

The equivalence is given by letting $\rho = r|_{G_{F,S}}$ and $r(c) = (B,-\chi (c))\jmath$. Conjugating $r$ by $M \in \operatorname {GL}_n(R)$ takes $B$ to $MB ^{t}{}{M}$. Note that the matrix $B$ defines an isomorphism $\chi \otimes \rho ^{\vee } \overset {\sim }{\rightarrow } \rho ^{c}$.

We embed the group $\mu _2^{{d}}$ in $\operatorname {GL}_n({{\mathcal {O}}})$ as block diagonal matrices, the $i$th block being of size $\dim _k \bar {\rho }_i$. We assume that the global deformation problem ${{\mathcal {S}}}$ has the property that each local deformation problem ${{\mathcal {D}}}_v \subset {{\mathcal {D}}}_v^{\square }$ is invariant under conjugation by $\mu _2^{{d}}$; this is the case for each of the local deformation problems recalled above. With this assumption, the group $\mu _2^{{d}}$ acts on the ring $R_{{{\mathcal {S}}}}^{\rm univ}$ by conjugation of the universal deformation and we have the following result.

Proposition 3.2

  1. (i) We have an equality $P_{{\mathcal {S}}} = (R_{{\mathcal {S}}}^{\text{univ}})^{\mu _2^{{d}}}$.

  2. (ii) Let ${{\mathfrak p}} \subset R_{{\mathcal {S}}}^{\text{univ}}$ be a prime ideal and let ${{\mathfrak q}} = {{\mathfrak p}} \cap P_{{\mathcal {S}}}$. Let $E = \operatorname {Frac} R_{{\mathcal {S}}}^{\text{univ}} / {{\mathfrak p}}$ and suppose that the associated representation $\rho _{{\mathfrak p}} = r_{{\mathfrak p}}|_{G_{F, S}}\otimes _A E : G_{F, S} \to \operatorname {GL}_n(E)$ is absolutely irreducible. Then $P_{{\mathcal {S}}} \to R_{{\mathcal {S}}}^{\text{univ}}$ is étale at ${{\mathfrak q}}$ and $\mu _2^{{d}}$ acts transitively on the set of primes of $R_{{{\mathcal {S}}}}^{\text{univ}}$ above ${{\mathfrak q}}$.

We first establish a preliminary lemma, before proving the proposition.

Lemma 3.3 Let $R = R_{{{\mathcal {S}}}}^{\text{univ}}/({{\mathfrak m}}_{P_{{{\mathcal {S}}}}})$ and let $r: G_{F^{+},S}\rightarrow {{\mathcal {G}}}_n(R)$ be a representative of the specialization of the universal deformation. Then, after possibly conjugating by an element of $1+M_n({{\mathfrak m}}_R)$, $r|_{G_{F,S}}$ has (block) diagonal entries given by $\bar {\rho }_1,\ldots ,\bar {\rho }_{{d}}$, and the matrix $B$ defined above is equal to $\bar {B}$. (Note we are not asserting that the off-diagonal blocks of $r|_{G_{F, S}}$ are zero.)

Proof. We let $\bar {e}_1,\bar {e}_2,\ldots ,\bar {e}_{{d}} \in M_n(k)$ denote the standard idempotents decomposing $\bar {r}|_{G_{F,S}}$ into the block diagonal pieces $\bar {\rho }_1,\ldots ,\bar {\rho }_{{d}}$. We let ${{\mathcal {A}}} \subset M_n(R)$ denote the image of $R[G_{F,S}]$ under $r$. The idempotents $\bar {e}_i$ lift to orthogonal idempotents $e_i$ in ${{\mathcal {A}}}$ with $e_1 + \cdots + e_{{d}} = 1$ and, after conjugating by an element of $1+M_n({{\mathfrak m}}_R)$, we can assume that the $e_i$ are again the standard idempotents on $R^{n}$. Moreover, applying the first case of the proof of [Reference Bellaïche and ChenevierBC09, Lemma 1.8.2], we can (and do) choose the $e_i$ so that they are fixed by the anti-involution $\star : {{\mathcal {A}}} \to {{\mathcal {A}}}$ given by the formula $M \mapsto B (^{t}{}{M}) B^{-1}$. This implies that the matrix $B$ is block diagonal. We have $e_i{{\mathcal {A}}}e_i = M_{n_i}(R)$ (see [Reference Bellaïche and ChenevierBC09, Lemma 1.4.3] and [Reference ChenevierChe14, Theorem 2.22]) and,for each $i \ne j$, we have $e_i{{\mathcal {A}}}e_j = M_{n_i,n_j}({{\mathcal {A}}}_{i,j})$, where ${{\mathcal {A}}}_{i,j} \subset R$ is an ideal [Reference Bellaïche and ChenevierBC09, Proposition 1.3.8].

Since $\det \circ \, r = \det \circ \, \bar {r}$, Proposition 2.5 shows that $\sum _{i \ne j}{{\mathcal {A}}}_{i,j}{{\mathcal {A}}}_{j,i} = 0$. This implies that for each $i$ the map

\[ R[G_{F,S}] \to M_{n_i}(R) \]

given by

\[ x \mapsto e_i r(x) e_i \]

is an algebra homomorphism and we get an $R$-valued lift of $\bar {\rho }_i$. By the uniqueness assertion in Proposition 2.5, the determinant of this lift is equal to $\det \circ \bar {\rho }_i$. Since $\bar {\rho }_i$ is absolutely irreducible, it follows from [Reference ChenevierChe14, Theorem 2.22] that, after conjugating by a block diagonal matrix in $1+M_n({{\mathfrak m}}_R)$, we can assume that the map

\[ x \mapsto e_i r(x) e_i \]

is induced by $\bar {\rho }_i$, which is the desired statement.

Finally, we consider the matrix $B$. We have already shown that $B$ is block diagonal. For $1\le i \le {{d}}$, we denote the corresponding block of $B$ by $B_i$. It lifts a block $\bar {B}_i$ of $\bar {B}$. By Schur's lemma, we have $B_i = \beta _i\bar {B}_i$ for some scalars $\beta _i \in 1+{{\mathfrak m}}_R$. Since $2$ is invertible in $R$, we can find $\lambda _i \in 1+{{\mathfrak m}}_R$ with $\lambda _i^{2} = \beta _i^{-1}$. Conjugating $r$ by the diagonal matrix with $\lambda _i$ in the $i$th block puts $r$ into the desired form.

Proof of Proposition 3.2 We begin by proving the first part. We again let $R = R_{{{\mathcal {S}}}}^{\rm univ}/({{\mathfrak m}}_{P_{{{\mathcal {S}}}}})$. By Nakayama's lemma, it suffices to show that $R^{\mu _2^{{d}}} = k$. Indeed, the natural map $(R_{{{\mathcal {S}}}}^{\rm univ})^{\mu _2^{{d}}}/{{\mathfrak m}}_{P_{{{\mathcal {S}}}}}(R_{{{\mathcal {S}}}}^{\rm univ})^{\mu _2^{{d}}} \to R^{\mu _2^{{d}}}$ is injective (i.e. $({{\mathfrak m}}_{P_{{{\mathcal {S}}}}}R_{{{\mathcal {S}}}}^{\rm univ})^{\mu _2^{{d}}} = {{\mathfrak m}}_{P_{{{\mathcal {S}}}}}(R_{{{\mathcal {S}}}}^{\rm univ})^{\mu _2^{{d}}}$), since if $\sum _i m_i x_i$ is ${\mu _2^{{d}}}$-invariant, with $m_i \in {{\mathfrak m}}_{P_{{{\mathcal {S}}}}}$ and $x_i \in R_{{{\mathcal {S}}}}^{\rm univ}$, we have $\sum _i m_i x_i = ({1}/{2^{d}})\sum _i m_i\sum _{\sigma \in \mu _2^{d}} \sigma x_i$, which is an element of ${{\mathfrak m}}_{P_{{{\mathcal {S}}}}}(R_{{{\mathcal {S}}}}^{\rm univ})^{\mu _2^{{d}}}$. Let $r : G_{F^{+}, S} \to {{\mathcal {G}}}_n(R)$ be a representative of the specialization of the universal deformation satisfying the conclusion of Lemma 3.3. Then $R$ is a finite $k$-algebra and is generated as a $k$-algebra by the matrix entries of $r$ and hence the matrix entries of $\rho = r|_{G_{F, S}}$ (because $B = \bar {B}$). We recall the ideals ${{\mathcal {A}}}_{i,j}\subset R$ appearing in the proof of Lemma 3.3, which are generated by the block $(i,j)$ matrix entries of $\rho$. The conjugate self-duality of $\rho$ is given by ${}^{t} \rho (\delta ) = \chi (\delta )\bar {B}^{-1} \rho ((\delta ^{c})^{-1}) \bar {B}$, $\delta \in G_{F,S}$. Since $\bar {B}$ is block diagonal, we deduce that ${{\mathcal {A}}}_{i,j}= {{\mathcal {A}}}_{j,i}$. Since $\sum _{i \ne j}{{\mathcal {A}}}_{i,j}{{\mathcal {A}}}_{j,i} = 0$, we see that ${{\mathcal {A}}}_{i,j}^{2} = 0$ for $i \ne j$. We deduce that $R$ is generated as a $k$-module by $1 \in R$ and products of the form

\[ a_{{{\mathcal{P}}}} = \prod_{(i,j) \in {{\mathcal{P}}}} a_{i,j}, \]

where $\emptyset \ne {{\mathcal {P}}} \subset \{(i,j): 1\le i < j \le {{d}}\}$ and $a_{i,j} \in {{\mathcal {A}}}_{i,j}$ has action of $\mu _2^{{d}}$ given by $((-1)^{\alpha _1},\ldots , (-1)^{\alpha _{{d}}})a_{i,j} = (-1)^{\alpha _i+\alpha _j}a_{i,j}$. Suppose that the action of $\mu _2^{{d}}$ on $a_{{{\mathcal {P}}}}$ is trivial. Then, for each $1 \le i \le {{d}}$, $i$ appears in an even number of elements of ${{\mathcal {P}}}$. A product $a'_{j_1,j_2} = a_{1,j_1}a_{1,j_2}$ lies in ${{\mathcal {A}}}_{j_1,j_2}$ and the action of $\mu _2^{{d}}$ is given by $((-1)^{\alpha _1},\ldots ,(-1)^{\alpha _{{d}}})a'_{j_1,j_2} = (-1)^{\alpha _{j_1}+\alpha _{j_2}}a'_{j_1,j_2}$. Since $1$ appears in an even number of elements of ${{\mathcal {P}}}$, we can ‘pair off’ these elements and rewrite $a_{{\mathcal {P}}}$ as a product

\[ a_{{{\mathcal{P}}}'} = \prod_{(i,j) \in {{\mathcal{P}}}'} a'_{i,j}, \]

where ${{\mathcal {P}}}' \subset \{(i,j): 2\le i < j \le {{d}}\}$ and the action of $\mu _2^{{d}}$ on $a'_{i,j}$ is given by the same formula as for $a_{i,j}$. Continuing in this manner, we deduce that $a_{{{\mathcal {P}}}}$ is the product of an even number of elements of ${{\mathcal {A}}}_{{{d}}-1, {{d}}}$ and thus equals $0$ since ${{\mathcal {A}}}_{{{d}}-1, {{d}}}^{2} = 0$.

The invariant subring $R^{\mu _2^{{d}}}$ is equal to the $k$-submodule of $R$ generated by $\sum _{\sigma \in \mu _2^{{d}}} \sigma x$, where $x$ runs over a set of $k$-module generators of $R$ (since $2$ is invertible in $k$). It follows from the above calculation that $R^{\mu _2^{{d}}} = k$.

We now prove the second part. The diagonally embedded subgroup $\mu _2 \subseteq \mu _2^{{d}}$ acts trivially on $R_{{{\mathcal {S}}}}^{\rm univ}$, so we have an induced action of $\mu _2^{{d}}/\mu _2$. The first part together with [Sta17, Tag 0BRI] implies that $\mu _2^{{d}} / \mu _2$ acts transitively on the set of primes of $R_{{{\mathcal {S}}}}^{\rm univ}$ above ${{\mathfrak q}}$. Let $R = R_{{{\mathcal {S}}}}^{\rm univ}/{{\mathfrak p}}$ and let $r_{{{\mathfrak p}}} \colon G_{F^{+},S} \rightarrow {{\mathcal {G}}}_n(R)$ be a representative of the specialization of the universal deformation. By [Sta17, Tag 0BST], to finish the proof if will be enough to show that if $\sigma \in \mu _2^{{d}}$, $\sigma ({{\mathfrak p}}) = {{\mathfrak p}}$, and $\sigma$ acts as the identity on $R$, then $\sigma \in \mu _2$.

If $\sigma \in \mu _2^{{d}}$ corresponds to the block diagonal matrix $g\in \operatorname {GL}_n({{\mathcal {O}}})$, then these conditions imply that $r_{{{\mathfrak p}}}$ and $g r_{{{\mathfrak p}}} g^{-1}$ are conjugate by an element $\gamma \in 1+M_n({{\mathfrak m}}_R)$. Since $r_{{{\mathfrak p}}}|_{G_{F,S}}\otimes E = \rho _{{{\mathfrak p}}}$ is absolutely irreducible, this implies that $g\gamma ^{-1}$ is scalar and so $g$ must also be scalar as $l>2$; hence, $g \in \mu _2$. This completes the proof.

For each partition $\{1, \ldots , {{d}}\} = {{\mathcal {P}}}_1 \sqcup {{\mathcal {P}}}_2$ with ${{\mathcal {P}}}_1, {{\mathcal {P}}}_2$ both non-empty, Proposition 2.5 gives an ideal $I_{({{\mathcal {P}}}_1, {{\mathcal {P}}}_2)} \subset P_{{\mathcal {S}}}$ cutting out the locus where the determinant $\det r|_{G_{F, S}}$ is $({{\mathcal {P}}}_1, {{\mathcal {P}}}_2)$-reducible. We write $I_{{\mathcal {S}}}^{\rm red} = \prod _{({{\mathcal {P}}}_1, {{\mathcal {P}}}_2)} I_{({{\mathcal {P}}}_1, {{\mathcal {P}}}_2)}$, an ideal of $P_{{\mathcal {S}}}$.

Lemma 3.4 Let ${{\mathfrak p}} \subset R_{{\mathcal {S}}}^{\text {univ}}$ be a prime ideal and let ${{\mathfrak q}} = {{\mathfrak p}} \cap P_{{{\mathcal {S}}}}$. Let $A = R_{{{\mathcal {S}}}}^{\text {univ}} / {{\mathfrak p}}$, $E = \operatorname {Frac} A$. Then $\rho _{{\mathfrak p}} = r_{{\mathfrak p}}|_{G_{F, S}}\otimes _A E$ is absolutely irreducible if and only if $I_{{\mathcal {S}}}^{\rm red} \not \subset {{\mathfrak q}}$.

Proof. If $I_{{\mathcal {S}}}^{\rm red} \subset {{\mathfrak q}}$, then $I_{({{\mathcal {P}}}_1, {{\mathcal {P}}}_2)} \subset {{\mathfrak q}}$ for some proper partition $({{\mathcal {P}}}_1, {{\mathcal {P}}}_2)$. Then Proposition 2.5 implies that $\det r_{{\mathfrak p}}$ admits a decomposition $\det \circ r_{{\mathfrak p}}|_{G_{F,S}} = D_1 D_2$ for two determinants $D_i : A[G_{F, S}] \to M_{n_i}(A)$. Then [Reference ChenevierChe14, Corollary 2.13] implies that $\rho _{{\mathfrak p}}$ is not absolutely irreducible.

Suppose conversely that $\rho _{{\mathfrak p}}$ is not absolutely irreducible. Let $J_{({{\mathcal {P}}}_1, {{\mathcal {P}}}_2)}$ denote the image of $I_{({{\mathcal {P}}}_1, {{\mathcal {P}}}_2)}$ in $A$. We must show that some $J_{({{\mathcal {P}}}_1, {{\mathcal {P}}}_2)}$ is zero. Let ${{\mathcal {A}}}$ denote the image of $A[G_{F, S}]$ in $M_n(A)$ under $r_{{\mathfrak p}}|_{G_{F, S}}$. According to [Reference Bellaïche and ChenevierBC09, Theorem 1.4.4], we can assume that ${{\mathcal {A}}}$ has the form

(2)\begin{equation} {{\mathcal{A}}} = \begin{pmatrix} M_{n_1}(A) & M_{n_1,n_2}(\mathcal{A}_{1,2}) & \cdots & M_{n_1,n_{{d}}}(\mathcal{A}_{1,{{d}}})\\ M_{n_2,n_1}(\mathcal{A}_{2,1}) & M_{n_2}(A) & \cdots & M_{n_2,n_{{d}}}(\mathcal{A}_{2,{{d}}})\\ \vdots & \vdots & \ddots & \vdots \\ M_{n_{{d}},n_1}(\mathcal{A}_{{{d}},1}) & M_{n_{{d}},n_2}(\mathcal{A}_{{{d}},2}) & \cdots & M_{n_{{d}}}(A) \end{pmatrix}\!, \end{equation}

where each $\mathcal {A}_{i, j}$ is a fractional ideal of $E$. Consequently, ${{\mathcal {A}}} \otimes _A E$ has the form

(3)\begin{equation} {{\mathcal{A}}} \otimes_A E= \begin{pmatrix} M_{n_1}(E) & M_{n_1,n_2}(\mathcal{E}_{1,2}) & \cdots & M_{n_1,n_{{d}}}(\mathcal{E}_{1,{{d}}})\\ M_{n_2,n_1}(\mathcal{E}_{2,1}) & M_{n_2}(E) & \cdots & M_{n_2,n_{{d}}}(\mathcal{E}_{2,{{d}}})\\ \vdots & \vdots & \ddots & \vdots \\ M_{n_{{d}},n_1}(\mathcal{E}_{{{d}},1}) & M_{n_{{d}},n_2}(\mathcal{E}_{{{d}},2}) & \cdots & M_{n_{{d}}}(E) \end{pmatrix}\!, \end{equation}

where each $\mathcal {E}_{i, j} = \mathcal {A}_{i, j} \otimes _A E$ equals either $E$ or $0$. Let $f_i \in M_n(E)$ denote the matrix with 1 in the $(i, i)$th entry and 0 everywhere else. If $\rho _{{\mathfrak p}}$ is not absolutely irreducible, then ${{\mathcal {A}}} \otimes _A E$ is a proper subspace of $M_n(E)$, so there exists $i$ such that $({{\mathcal {A}}} \otimes _A E) f_i \subset M_n(E) f_i$ is a proper subspace. Since $M_n(E) f_i$ is isomorphic as an $M_n(E)$-module to the tautological representation $E^{n}$, this implies that the ${{\mathcal {A}}} \otimes _A E$-module $E^{n}$ admits a proper invariant subspace. After permuting the diagonal blocks, we can assume that this subspace is $E^{n_1 + \cdots + n_s}$ for some $s < {{d}}$ (included as the subspace of $E^{n}$ where the last $n_{s+1} + \cdots + n_{{d}}$ entries are zero). Otherwise said, the spaces $\mathcal {E}_{i, j}$ for $i > s$, $j \leq s$ are zero. If ${{\mathcal {P}}}_1 = \{ 1, \ldots , s \}$ and ${{\mathcal {P}}}_2 = \{ s+1, \ldots , {d}\}$, then this implies that ${{\mathcal {J}}}_{({{\mathcal {P}}}_1, {{\mathcal {P}}}_2)} \otimes _A E = 0$ and hence (as $A$ is a domain) ${{\mathcal {J}}}_{({{\mathcal {P}}}_1, {{\mathcal {P}}}_2)} = 0$. This completes the proof.

Lemma 3.5 Let ${{\mathfrak p}} \subset R_{{\mathcal {S}}}^{\text {univ}}$ be a prime ideal, $A = R_{{\mathcal {S}}}^{\text {univ}} / {{\mathfrak p}}$, $E = \operatorname {Frac} A$. Then $r_{{\mathfrak p}}\otimes _A E$ is Schur and, if $r_{{\mathfrak p}}|_{G_{F, S}} \otimes _A E$ is not absolutely irreducible, then $r_{{\mathfrak p}}$ is equivalent (i.e. conjugate by an element in $1+ M_n({{\mathfrak m}}_A)$) to a type-${{\mathcal {S}}}$ lifting of the form $r_{{\mathfrak p}} = r_1 \oplus r_2$, where $r_i : G_{F^{+}, S} \to {{\mathcal {G}}}_{m_i}(A)$ and $m_1 m_2 \neq 0$.

Proof. We argue, as in the proof of Lemma 3.4, using the image ${{\mathcal {A}}} \subset M_n(A)$ of $A[G_{F, S}]$, which is a generalized matrix algebra. Suppose that we are given $G_{F, S}$-invariant subspaces $E^{n} \supset W_1 \supset W_2$ such that $W_2$ and $E^{n} / W_1$ are irreducible. We can assume that ${{\mathcal {A}}}$ has the form (2) and that this decomposition is block upper triangular (perhaps with respect to a coarser partition than $n = n_1 + \cdots + n_{{d}}$) and moreover than the first block corresponds to $W_2$, while the last block corresponds to $E^{n} / W_1$. In particular, $W_2$ and $E^{n} / W_1$ are even absolutely irreducible. Note that there can be no isomorphism $W_2^{c \vee }(\nu \circ r_{{\mathfrak p}}) \cong E^{n} / W_1$; if there was, then it would imply an identity of $A$-valued determinants, which we could reduce modulo ${{\mathfrak m}}_A$ to obtain an identity $\{ \rho _{i} \} = \{ \rho _j \}$ of sets of irreducible constituents of $\bar {r}|_{G_{F, S}}$. Since these appear with multiplicity 1, this is impossible. This all shows that $r_{{\mathfrak p}} \otimes _A E$ is necessarily Schur.

Now suppose that $r_{{\mathfrak p}}|_{G_{F, S}}\otimes _A E$ is not absolutely irreducible. After permuting the diagonal blocks of $\bar {r}$, we can assume that there is some $1 \leq m \leq {d}$ such that ${{\mathcal {A}}}_{i, j} = 0$ for $i > m$, $j \leq m$. The existence of the conjugate self-duality of $r_{{\mathfrak p}}$ implies (cf. [Reference Bellaïche and ChenevierBC09, Lemma 1.8.5]) that ${{\mathcal {A}}}_{j, i} = 0$ in the same range, giving a decomposition $r_{{\mathfrak p}}|_{G_{F, S}} = \rho _1 \oplus \rho _2$ of representations over $A$. Since $r_{{\mathfrak p}} \otimes _A E$ is Schur, the conjugate self-duality of $r_{{\mathfrak p}}$ must make $\rho _1$ and $\rho _2$ orthogonal, showing that $r_{{\mathfrak p}}$ itself decomposes as $r_{{\mathfrak p}} = r_1 \oplus r_2$.

3.3 Dimension bounds

We now suppose that $S$ admits a decomposition $S = S_l \sqcup S(B) \sqcup R \sqcup S_a$, where:

  1. for each $v \in S(B) \cup R$, $q_v \equiv 1 \text { mod }l$ and $\bar {r}|_{G_{F_{{\widetilde {v}}}}}$ is trivial;

  2. for each $v \in S_a$, $q_v \not \equiv 1 \text { mod }l$, $\bar {r}|_{G_{F_{{\widetilde {v}}}}}$ is unramified, and $\bar {r}|_{G_{F_{{\widetilde {v}}}}}$ is scalar. (Then any lifting of $\bar {r}|_{G_{F_{{\widetilde {v}}}}}$ is unramified.)

We consider the global deformation problem

\[ {{\mathcal{S}}} = ( F / F^{+}, S, \tilde{S}, \Lambda, \bar{r}, \chi, \{ {{\mathcal{D}}}_v^{\triangle} \}_{v \in S_l} \cup \{ {{\mathcal{D}}}_v^{\rm St} \}_{v \in S(B)} \cup \{ {{\mathcal{D}}}_v^{1} \}_{v \in R} \cup \{ {{\mathcal{D}}}_v^{\square} \}_{v \in S_a} ), \]

where $\bar {r}$ is assumed to be Schur. We define quantities $d_{F, 0} = d_0$ and $d_{F, l} = d_l$ as follows. Let $\Delta$ denote the Galois group of the maximal abelian pro-$l$ extension of $F$ which is unramified outside $l$, and let $\Delta _0$ denote the Galois group of the maximal abelian pro-$l$ extension of $F$ which is unramified outside $l$ and in which each place of $S(B)$ splits completely. We set

\[ d_0 = \dim_{{{\mathbb{Q}}}_l} \ker( \Delta[1/l] \to \Delta_0[1/l] )^{c = -1} \]


\[ d_l = \inf_{v \in S_l} [F^{+}_v : {{\mathbb{Q}}}_l]. \]

Lemma 3.6 Suppose that $d_l > n(n-1)/2 + 1$. Let $A \in {{\mathcal {C}}}_\Lambda$ be a finite $\Lambda$-algebra and let $r : G_{F^{+}, S} \to {{\mathcal {G}}}_n(A)$ be a lifting of $\bar {r}$ of type ${{\mathcal {S}}}$. Then $\dim A / (I_{{\mathcal {S}}}^{\rm red}, \lambda ) \leq n[F^{+} : {{\mathbb {Q}}}] - d_0$.

Proof. We can assume without loss of generality that $A = A / (I_{{\mathcal {S}}}^{\rm red}, \lambda )$ and must show that $\dim A \leq [F^{+} : {{\mathbb {Q}}}] - d_0$. Since $A$ is Noetherian and we are interested only in dimension, we can assume moreover that $A$ is integral. Let $E = \operatorname {Frac}(A)$. Then (Lemma 3.5) we can find a non-trivial partition $n = n_1 + n_2$ and homomorphisms $r_i : G_{F^{+}, S} \to {{\mathcal {G}}}_{n_i}(A)$ ($i = 1, 2$) such that $r = r_1 \oplus r_2$.

Let $\bar {E}$ be a choice of algebraic closure of $E$. Our condition on $d_l$ means that we can appeal to [Reference ThorneTho15, Corollary 3.12] (characterization of $A$-valued points of ${{\mathcal {D}}}_v^{\triangle }$ for each $v \in S_l$). This result implies the existence for each $v \in S_l$ of an increasing filtration

\[ 0 \subset \operatorname{Fil}^{1}_v \subset \operatorname{Fil}^{2}_v \subset \cdots \subset \operatorname{Fil}^{n}_v = \bar{E}^{n} \]

of $r|_{G_{F_{{\widetilde {v}}}}} \otimes _A \bar {E}$ by $G_{F_{{\widetilde {v}}}}$-invariant subspaces such that each $\operatorname {Fil}^{i}_v / \operatorname {Fil}^{i-1}_v$ is one dimensional, and the character of $I^{{\text ab}}_{F_{{\widetilde {v}}}}(l)$ acting on this space is given by composing the universal character $\psi _v^{i} : I^{{\text ab}}_{F_{{\widetilde {v}}}}(l) \to \Lambda _v^{\times }$ with the homomorphism

\[ \Lambda_v \to \Lambda \to A \to \bar{E}. \]

The direct sum decomposition of $r$ leads to a decomposition $r|_{G_{F_{{\widetilde {v}}}}} = r_1|_{G_{F_{{\widetilde {v}}}}} \oplus r_2|_{G_{F_{{\widetilde {v}}}}}$. Let $F_v^{i} = \operatorname {Fil}_v^{i} \cap r_1|_{G_{F_{{\widetilde {v}}}}} \otimes _A \bar {E}$ and $G_v^{i} = \operatorname {Fil}_v^{i} \cap r_2|_{G_{F_{{\widetilde {v}}}}} \otimes _A \bar {E}$. Then $F_v^{\bullet }$ and $G_v^{\bullet }$ are increasing filtrations of $\bar {E}^{n_1}$ and $\bar {E}^{n_2}$, respectively, with graded pieces of dimension at most 1. We write $\sigma _v$ for the bijection

\[ \sigma_v : \{ 1, \ldots, n_1 \} \sqcup \{ 1, \ldots, n_2 \} \to \{1, \ldots, n \}, \]

which is increasing on $\{ 1, \ldots , n_1 \}$ and $\{ 1, \ldots , n_2 \}$ and which has the property that $\sigma _v( \{ 1, \ldots , n_1 \} )$ is the set of $i \in \{1, \ldots , n\}$ such that the graded piece $F_v^{i} / F_v^{i-1}$ is non-trivial.

Let $\Lambda _{v, 1}$ and $\Lambda _{v, 2}$ denote the analogues of the algebra $\Lambda _v$ in dimensions $n_1$ and $n_2$, respectively. The bijection $\sigma _v$ determines in an obvious way an isomorphism $\Lambda _{v, 1} \hat {\otimes } \Lambda _{v, 2} \cong \Lambda _v$. Applying again [Reference ThorneTho15, Corollary 3.12], we see that with this structure on