2.1 Notation related to global and local fields

We write $K^{\rm s}$ for a fixed choice of separable closure and $\Gamma _K := \operatorname {Gal}(K^{\rm s} / K)$ for the corresponding Galois group. For $S$ a finite set of places of $K$, we write $K_S$ for the maximal subextension of $K^{\rm s}$ unramified outside $S$, and $\Gamma _{K, S} := \operatorname {Gal}(K_S / K)$. If $v$ is a place of $K$, then $\Gamma _{K_v} = \operatorname {Gal}(K_v^{\rm s} / K_v)$ will denote the decomposition group, and $\Gamma _{K_v} \to \Gamma _K$ the homomorphism corresponding to a fixed choice of $K$-embedding $K^{\rm s} \hookrightarrow K_v^{\rm s}$. If $v \not \in S$, then $\operatorname {Frob}_v \in \Gamma _{K, S}$ denotes a choice of geometric Frobenius element at the place $v$. We will identify the set of places of $K$ with the set of closed points of $X$. For a place $v \in |X|$ we write $q_v = \# k(v) = \# ({\mathcal {O}}_{K_v} / \varpi _v {\mathcal {O}}_{K_v})$ for the size of the residue field at $v$. We write $| \cdot |_v$ for the norm on $K_v$, normalized so that $|\varpi _v|_v = q_v^{-1}$; then the product formula holds. We write $\widehat {{\mathcal {O}}}_{K} = \prod _{v \in X} {\mathcal {O}}_{K_v}$. We will write $W_{K_v}$ for the Weil group of the local field $K_v$.

2.2 Notation related to group schemes

If $G, H, \ldots$ are group schemes over a base $S$, then we use Gothic letters ${\mathfrak g}, {\mathfrak h}, \ldots$ to denote their Lie algebras, and $G_T, {\mathfrak g}_T, \ldots$ to denote the base changes of these objects relative to a scheme $T \to S$. If $G$ acts on an $S$-scheme $X$ and $x \in X(T)$, then we write $Z_G(x)$ or $Z_{G_T}(x)$ for the scheme-theoretic stabilizer of $x$; it is a group scheme over $T$. We denote the center of $G$ by $Z_G$. We say that a group scheme $G$ over $S$ is reductive if $G$ is smooth and affine with reductive (and therefore connected) geometric fibres.

2.3 Notation related to coefficient rings

When doing deformation theory, we will generally fix a prime $\ell$ and an algebraic closure $\overline {{\mathbb {Q}}}_\ell$ of ${\mathbb {Q}}_\ell$. A finite extension $E / {\mathbb {Q}}_\ell$ inside $\overline {{\mathbb {Q}}}_\ell$ will be called a coefficient field; when such a field $E$ has 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 $\varpi$ or $\varpi _E$ for a choice of uniformizer of ${\mathcal {O}}_E$. We write ${\mathcal {C}}_{\mathcal {O}}$ for the category of Artinian local ${\mathcal {O}}$-algebras with residue field $k$; if $A \in {\mathcal {C}}_{\mathcal {O}}$, then we write ${\mathfrak m}_A$ for its maximal ideal. Then $A$ comes with the data of an isomorphism $k \cong A / {\mathfrak m}_A$.

2.4 Dual groups

In this paper, we will view the dual group $\widehat {G}$ of $G$ as a split reductive group over $\mathbb {Z}$. Our definition of $\widehat {G}$ follows [Reference Böckle, Harris, Khare and ThorneBHKT19, § 2.1]. A prime $\ell$ is called a very good characteristic for ${\widehat {G}}$ if it satisfies the conditions in the table below for all the simple factors of $\widehat {G}$ (referring to the absolute root system types):

3.1 Setup for the general theory

Let ${\widehat {G}}$ be a split semisimple group over ${\mathbb {Z}}$. We fix a prime $\ell$ which is a very good characteristic for ${\widehat {G}}$, as well as a coefficient field $E \subset \overline {{\mathbb {Q}}}_\ell$. We also fix an absolutely irreducible representation $\overline {\sigma }$ of $\Gamma _{K,S}$. We recall the results of [Reference Böckle, Harris, Khare and ThorneBHKT19] on the deformation theory of representations of $\Gamma _{K,S}$ to ${\widehat {G}}$ with $\ell$-adic coefficients.

Let $A \in {\mathcal {C}}_{\mathcal {O}}$. We define liftings and deformations of ${\overline {\sigma }}$ over $A$ as in [Reference Böckle, Harris, Khare and ThorneBHKT19, § 5.1], and let $\operatorname {Def}_{{\overline {\sigma }}} : {\mathcal {C}}_{\mathcal {O}} \to \operatorname {Sets}$ be the functor that associates to $A \in {\mathcal {C}}_{\mathcal {O}}$ the set of deformations of ${\overline {\sigma }}$ over $A$. Then we have the following proposition.

3.2 Local deformation conditions

So far we have considered deformations of ${\overline {\sigma }}$ with no restriction on the points in $S$. We now introduce local deformation conditions, following the discussion in [Reference Clozel, Harris and TaylorCHT08, § 2.2], and for general groups in [Reference PatrikisPat16, § 3.2], with notation as in [Reference Khare and ThorneKT17, § 4]. For each $v \in S$ let ${\Gamma _{K_v}} \subset \Gamma _{K,S}$ be a decomposition group at $v$, and let ${\overline {\sigma }}_v$ denote the restriction of ${\overline {\sigma }}$ to ${\Gamma _{K_v}}$. Let ${\mathcal {D}}^\square _{{\overline {\sigma }},v}$ denote the functor on ${\mathcal {C}}_{\mathcal {O}}$ which to $A \in \mathcal{C}_{\mathcal{O}}$ associates the set of liftings of ${\overline{\sigma}}$ to A; then ${\mathcal {D}}^\square _{{\overline {\sigma }},v}$ is represented by a ring $R^\square _{{\overline {\sigma }},v}$. As in [Reference Khare and ThorneKT17, Definition 4.1], we define a local deformation problem for ${\overline {\sigma }}_v$ to be a subfunctor ${\mathcal {D}}_v \subset {\mathcal {D}}^\square _{{\overline {\sigma }},v}$ such that:

• • ${\mathcal {D}}_v$ is represented by a quotient of $R^\square _{{\overline {\sigma }},v}$; and

• • for any $A \in {\mathcal {O}}$, if $\sigma \in {\mathcal {D}}_v(A)$ and $\alpha \in \ker [{\widehat {G}}(A) \rightarrow {\widehat {G}}(k)]$ then ${\rm ad}(\alpha )(\sigma ) \in {\mathcal {D}}_v(A)$.

To each local deformation problem ${\mathcal {D}}_v$ we attach a $k$-subspace $L_v \subset H^1(G_{K_v}, {\rm ad}({\overline {\sigma }}))$ as in [Reference PatrikisPat16, § 3.2].

Example 3.5 The deformation problem ${\mathcal {D}}_v$ is unrestricted if ${\mathcal {D}}_v = {\mathcal {D}}^\square _{{\overline {\sigma }},v}$. In this case we have $L_v = H^1({\Gamma _{K_v}}, {\rm ad}({\overline {\sigma }}))$.

Recall that for all $v \in S$, Tate's local Euler characteristic formula for any $k[{\Gamma _{K_v}}]$-module $M$ reduces to

\[ h^0({\Gamma_{K_v}},M) - h^1({\Gamma_{K_v}},M) + h^2({\Gamma_{K_v}},M) = 0 \]

because $\ell \neq p$. Hence, if $H^2({\Gamma _{K_v}}, {\rm ad}({\overline {\sigma }})) = 0$ then the unrestricted deformation problem ${\mathcal {D}}_v$ is balanced.

Note that local duality implies that

(3.1)\begin{equation} H^2({\Gamma_{K_v}}, {\rm ad}({\overline{\sigma}})) = 0 \Leftrightarrow H^0({\Gamma_{K_v}}, {\rm ad}({\overline{\sigma}})^\vee(1)) = 0. \end{equation}

3.3 Selmer groups

Given a global deformation problem ${\mathcal {D}}$, we define

(3.2)\begin{equation} H^1_{{\mathcal{D}}}(\Gamma_{K,S},{\rm ad}({\overline{\sigma}})) := \ker \bigg[H^1(\Gamma_{K,S},{\rm ad}({\overline{\sigma}})) \rightarrow \bigoplus_{v \in S} H^1({\Gamma_{K_v}}, {\rm ad}({\overline{\sigma}}))/L_v \bigg], \end{equation}

and let $h^1_{{\mathcal {D}}}(\Gamma _{K,S},{\rm ad}({\overline {\sigma }}) ) := \dim _k H^1_{{\mathcal {D}}}(\Gamma _{K,S},{\rm ad}({\overline {\sigma }}))$.

With $L_v$ as above, let $L_v^\perp \subset H^1({\Gamma _{K_v}}, {\rm ad}({\overline {\sigma }})^{\vee }(1))$ denote its orthogonal complement under the local duality pairing. We define

\[ H^1_{{\mathcal{D}}^{\perp}}(\Gamma_{K,S},{\rm ad}({\overline{\sigma}})^{\vee}(1)) := \ker \bigg[H^1(\Gamma_{K,S},{\rm ad}({\overline{\sigma}})^{\vee}(1)) \rightarrow \bigoplus_{v \in S} H^1({\Gamma_{K_v}}, {\rm ad}({\overline{\sigma}})^{\vee}(1))/L_v^{\perp} \bigg], \]

and let $h^1_{{\mathcal {D}}^{\perp }}(\Gamma _{K,S},{\rm ad}({\overline {\sigma }})^{\vee } (1)) := \dim _k H^1_{{\mathcal {D}}^{\perp }}(\Gamma _{K,S},{\rm ad}({\overline {\sigma }})^{\vee }(1))$.

3.4 Abundant subgroups

We recall two definitions from [Reference Böckle, Harris, Khare and ThorneBHKT19].

Lemma 3.8 Suppose ${\widehat {G}}$ is simple and split over $k$. Let ${\widehat {G}}^{\rm sc}$ denote the simply connected cover of ${\widehat {G}}$, and let ${\widehat {G}}(k)^+$ denote the image of ${\widehat {G}}^{\rm sc}(k)$ in ${\widehat {G}}(k)$. Let $H$ be a group with ${\widehat {G}}^+(k)\subset H\subset {\widehat {G}}(k)$ and suppose that ($k$, type of ${\widehat {G}}$) is not in the following list:

\[ \{(\mathbb{F}_3,A_1), (\mathbb{F}_5,A_1) \}\cup \{(\mathbb{F}_q,C_n)\mid q\in\{3,5,9\},n\ge2\}. \]

Then there exists a finite field $k_1\supset k$ such that for all finite fields $k'\supset k_1$ the group $H$ is ${\widehat {G}}$-abundant over $k'$.

Proof. We will prove the following claims for $(k,\mathrm {type\, of\, }{\widehat {G}})$ not in the above list:

1. (i) The adjoint module $\widehat {{\mathfrak g}}_{k}$ is self-dual as a $k[H]$-representation,

2. (ii) The groups $H^0(H, \widehat {{\mathfrak g}}_{k})$, $H^1(H, \widehat {{\mathfrak g}}_{k})$, and $H^1(H, k)$ all vanish.

3. (iii) The action of $H$ on $\widehat {{\mathfrak g}}_{k}$ is absolutely irreducible.

4. (iv) The group $H$ contains strongly regular semisimple elements $h$, i.e. $h$ such that $Z_{{\widehat {G}}}(h)$ is connected.

Let us first show that the claims suffice to prove the lemma. Indeed, condition (a) is obvious, condition (b) is the same as (ii) (given (i)), while condition (c) is automatic if $k_1$ is chosen to split all of the finitely many $k$-rational tori $Z_{{\widehat {G}}}(h)^\circ$ in ${\widehat {G}}$. Finally, given (iii), condition (d) comes down to the existence of strongly regular semisimple $h$, as in (iv), because $(\widehat {{\mathfrak g}}^\vee )^h$ always contains the Lie algebra of the centralizer of $h$.

It remains to establish the claim. Parts (i) to (iii) with many details can be found in [Reference Arias-de Reyna and BöckleAB, § 3], which, however, largely builds on further references, which we now briefly recall. We first observe that $\widehat {{\mathfrak g}}_{k}$ is an absolutely irreducible $k[H]$-representation, essentially by results of Hiss [Reference HissHis84] and Hogeweij [Reference HogeweijHog82]; see [Reference Arias-de Reyna and BöckleAB, Proposition 3.22]. Now because $\widehat {{\mathfrak g}}_{k}$ is absolutely irreducible, to prove self-duality it suffices to see that $\widehat {{\mathfrak g}}_{k}$ and $\widehat {{\mathfrak g}}_{k}^\vee$ have the same weights. But this is clear since with any root $\alpha$ of the Lie algebra $\widehat {{\mathfrak g}}_{k}$, $-\alpha$ is also a root, and both have multiplicity $1$. This settles (i) and (iii), and the first assertion of (ii).

Next, a largely classical result, completed by Tits, asserts that ${\widehat {G}}(k)^+$ is perfect, unless $(k,\mathrm {type\, of\, }{\widehat {G}})$ is $(\mathbb {F}_3,A_1)$; see [Reference Malle and TestermanMT11, Thm. 24.17]. Because ${\widehat {G}}$ is split and $l$ is of very good characteristic for ${\widehat {G}}$, there is only a single exception. The latter also implies that ${\widehat {G}}(k)/{\widehat {G}}(k)^+$ is of order prime to the characteristic $l$ of $k$. Hence $H^1(H, k)=\operatorname {Hom}(H, k)$ vanishes, and this shows the third part of (ii); see [Reference Arias-de Reyna and BöckleAB, Corollary 3.12].

We now turn to the remaining condition $H^1({\widehat {G}}(k), \widehat {{\mathfrak g}}_{k}) = 0$ from (ii). The most complete vanishing results for $H^1(H, \widehat {{\mathfrak g}}_{k})$ are due to Völklein with earlier work by Cline, Parshall, Scott in the A-D-E cases and by Hertzig (unpublished). We observe that $\operatorname {Lie}{\widehat {G}}$ has trivial center because $l$ is very good for ${\widehat {G}}$. The latter also excludes the case $(k,\mathrm {type\ of\ }{\widehat {G}})= (\mathbb {F}_2,A_1)$. The cases $(\mathbb {F}_5,A_1)$ and $(\mathbb {F}_q,C_n)$, $q\in \{3,5,9\}$, $n\ge 2$, for $(k,\mathrm {type\, of\, }{\widehat {G}})$ are ruled out by our list of exceptions. It follows from [Reference VölkleinVöl89, Theorem and Remarks] that $H^1({\widehat {G}}(k), \widehat {{\mathfrak g}}_{k}) = 0$.

Finally, we prove (iv). If ${\widehat {G}}$ is simply connected, then by [Reference Springer and SteinbergSS70, II.3.9] the group $Z_{{\widehat {G}}}(h)$ is connected for all regular semisimple $h\in H$, and we are done. We now focus on the case where ${\widehat {G}}$ of adjoint type, where we find a suitable $h$ using [Reference Feit and SeitzFS88]. The last paragraph explains how this gives (iv) for ${\widehat {G}}$ of any other type.

Suppose that ${\widehat {G}}$ is of adjoint type. Then by [Reference Feit and SeitzFS88, Theorem 3.1] there exists a maximal torus $T\subset {\widehat {G}}$ defined over $k$ such that the following statements hold:

1. (1) The group $T^+(k):={\widehat {G}}^+(k)\cap T(k)$ is cyclic. Let $h$ be a generator.

2. (2) One has $Z_{{\widehat {G}}(k)}(h)=T(k)$, and the order of $h$ is given in [Reference Feit and SeitzFS88, Table III].

3. (3) If ${\widehat {G}}$ is of type $A_n$, then $T$ is quotient of a totally anisotropic maximal torus of $\operatorname {GL}_{n+1}$ (over $k$) modulo its center, i.e. $T\cong \operatorname {Res}_{k^{(n+1)}}^k \mathbb {G}_m/\mathbb {G}_m$, the quotient of the Weil restriction of $\mathbb {G}_m$ from the unique degree-($n+1$) extension $k^{(n+1)}$ of $k$ to $k$ modulo $\mathbb {G}_m$.

Let $C:=Z_{{\widehat {G}}}(h)$, and let $\pi _0(C)=C/C^0$ be its component group scheme. It follows from [Reference SeitzSei83, 2.9 and its proof] that the identity component $C^0$ is a (commuting) product of Chevalley groups over $k$ and of the center of $C^0$, and so from (1) above the natural map $T\to C^0$ must be an isomorphism because any Chevalley group has $k$-points of order $l$. In particular, $h$ is regular semisimple.

Next we gather results on $\pi _0( C)$ that will allow us to identify it with the trivial group. By [Reference Springer and SteinbergSS70, Corollary II.4.4] there is an injective morphism of group schemes $\pi _0(C)\to Z({\widehat {G}}^{\rm sc})$ to the center $Z({\widehat {G}}^{\rm sc})$ of ${\widehat {G}}^{\rm sc}$. Because $l$ is good for ${\widehat {G}}$, the finite commutative group scheme $Z({\widehat {G}}^{\rm sc})$ is étale over $k$. The centers are completely known, and a list can be found, for instance, in [Reference Arias-de Reyna and BöckleAB, Table 1]. Using the Bruhat decomposition of ${\widehat {G}}$ over $\overline k$, and that $h$ is regular semisimple, it is not difficult to show that $C$ is a subgroup of the normalizer of $T$ in ${\widehat {G}}$ (over $\overline k$), and hence $\pi _0(C)$ is a subgroup of the Weyl group of $G$. Moreover, by [Reference Springer and SteinbergSS70, I.2.11] one has a short exact sequence

\[ 0\to T(k) \to C(k) \to \pi_0(C)(k)\to 0. \]

From (2) above we deduce that $T(k)\to C(k)$ is an isomorphism, and hence $\pi _0( C)(k)$ is trivial.

Now suppose the type is $B_n$, $C_n$, $D_n$ or $E_7$. Then $l$ cannot be $2$, and so $2$ divides the order of $k^\times$. It follows from the classification of the centers $Z({\widehat {G}}^{\rm sc})$, that any non-trivial subgroup scheme will contain an element of exact order $2$. But then $\pi _0( C)$ must be trivial, because we know already that $\pi _0( C) (k)$ is trivial.

The types $E_8$, $F_4$ and $G_2$ need not be considered, since here $Z({\widehat {G}}^{\rm sc})$ is trivial and hence ${\widehat {G}}$ itself is simply connected. Next we consider type $E_6$, so that $Z({\widehat {G}}^{\rm sc})\cong \mu _3$. Let $q=\#k$. If $q \equiv 1 \pmod 3$, we can argue as in the previous paragraph to deduce that $\pi _0( C)$ is trivial. If, on the other hand, $q \equiv -1 \pmod 3$, then from [Reference Feit and SeitzFS88, Table III] we find that

\[ \#T(k) = (q^2+q+1)(q^4-q^2+1)\equiv -1\pmod 3, \]

so that the order of $h$ is not divisible by $3$. It follows from [Reference Springer and SteinbergSS70, Corollary 4.6] that $\pi _0( C)$ is connected.

It remains to discuss the type $A_n$. For $n=1$, by (3) the element $h$ is given as a diagonal matrix $\operatorname {diag}(x,x^q)$ in $\operatorname {GL}_2/\mathbb {G}_m$ for some $x\in (k^{(2)} )^\times$ of order $q^2-1$, so that $h$ has order $q+1$. Because $C$ is spanned by $T$ and at most some Weyl group elements, we need to understand whether the unique non-trivial Weyl group element $w$ of ${\widehat {G}}=\operatorname {PGL}_2$ lies in $C$. The element $w$ exchanges the entries of $h$. If it would fix $h$, then $\operatorname {diag}(x,x^q)\equiv \operatorname {diag}(x^q,x)\pmod {\overline k^\times }$, so that there exists $a\in \overline k^\times$ with $ax=x^q$ and $ax^q=x$. This firstly implies $a\in \{\pm 1\}$, and then that the order of $x$ divides $2(q-1)$. Because $q+1$ divides the order of $x$, this could only happen for $q=2$, which is forbidden, since $l$ is assumed to be very good for ${\widehat {G}}=\operatorname {PGL}_2$.

Now suppose $n\ge 2$. Then $h$ is given by $\operatorname {diag}(x,x^q,\ldots,x^{q^n})\in \operatorname {GL}_{n+1}/\mathbb {G}_m(\overline k)$ for some $x\in (k^{(n+1)})^\times$ of order $q^{n+1}-1$. Let $w$ be in the Weyl group of $\operatorname {PGL}_n$, which we identify with $S_{n+1}$. Suppose that $h=w\circ h$, i.e. that $w$ is the image of an element of $C$. Because $\pi _0( C)\subset Z(\operatorname {PGL}_{n+1})\cong \mu _{n+1}$ we may assume that $w$ is cyclic of order dividing $n+1$. Let $r_1\ge \cdots r_s\ge 2$ be the lengths in the cycle decomposition of $w$, and let $r=\gcd (r_1,\ldots r_s)$, which is now a divisor of $n+1$. As in the case $n=1$, we find $a\in \overline k^\times$ of order dividing $r$ and such that $x^{q^{w(i)}}=ax^{q^i}$, or equivalently $x^{q^{w(i)}-q^i}=a$, for $i=0,\ldots,n$.

To get further, let $t$ be a large Zsygmondy prime for $(q,n+1)$ in the sense of [Reference Feit and SeitzFS88, Theorem 2.1], i.e. $t$ is a prime that divides $q^{n+1}-1$, that does not divide $q^m-1$ for $1\le m \le n$ and with either $t>n+1$ or $t^2|(q^{n+1}-1)$. We deduce that $t$ divides the order of $x^{q^{w(i)}-q^i}$ for any $i$ with $w(i)\neq i$, and it follows that $t$ divides $r$ and in turn $n+1$, unless $w$ is trivial. But then modulo the prime $t$ and using Fermat's little theorem we have

\[ q^{n+1}-1 = (q^t)^{(n+1)/t} -1\equiv q^{(n+1)/t} -1 \pmod t. \]

This contradicts that $t$ is a Zsygmondy prime for $(q,n+1)$. Hence $w$ must be trivial and $T=C$.

Finally, for a general ${\widehat {G}}$ consider the central isogeny ${\widehat {G}}\to {\widehat {G}}^{\rm ad}$ to the adjoint quotient ${\widehat {G}}^{\rm ad}$ of ${\widehat {G}}$. Let $H^{\rm ad}\subset {\widehat {G}}^{\rm ad}(k)$ be the image of $H$. Now the element $h$ in $H^{ad,+}(k)$ from (1) above clearly lifts to some $h'\in {\widehat {G}}^+(k)$ with $h'$ regular semisimple. Moreover, we have a left exact sequence $0\to Z({\widehat {G}})\to Z_{{\widehat {G}}}(h')\to Z_{{\widehat {G}}^{\rm ad}}(h)$. In it, $Z_{{\widehat {G}}^{\rm ad}}(h)$ is a maximal torus, $Z_{{\widehat {G}}}(h')$ contains a maximal torus, and the center $Z({\widehat {G}})$ lies in any maximal torus of ${\widehat {G}}$. Hence $Z_{{\widehat {G}}}(h')$ is a (maximal) torus, and thus connected.

3.5 Taylor–Wiles primes

Following [Reference Böckle, Harris, Khare and ThorneBHKT19, Definition 5.16], we define a Taylor–Wiles datum for $\overline {\sigma }$ to be a pair $(Q, \{ \varphi _v \}_{v \in Q})$, where the following statements hold:

• • $Q$ is a finite set of places $v$ of $K$ such that $\overline {\sigma }(\operatorname {Frob}_v)$ is regular semisimple and $q_v \equiv 1 \text { mod }\ell$

• • For each $v \in Q$, $\varphi _v : {\widehat {T}}_k \cong Z_{\widehat {G}}(\overline {\sigma }(\operatorname {Frob}_v))$ is a choice of inner isomorphism. In particular, this forces $Z_{\widehat {G}}(\overline {\sigma }(\operatorname {Frob}_v))$ to be connected.

For a global deformation problem ${\mathcal {D}} = (S, \{{\mathcal {D}}_v\}_{v \in S})$ and a Taylor–Wiles datum $(Q, \{ \varphi _v \}_{v \in Q})$ with $Q \cap S = \emptyset$, we define a new global deformation problem ${\mathcal {D}}_Q = (S \cup Q, ({\mathcal {D}}_Q)_v )$ with

\[ ({\mathcal{D}}_Q)_v = \begin{cases} {\mathcal{D}}_v & v \in S, \\ {\mathcal{D}}_{{\overline{\sigma}}, v}^{\square} & v \in Q. \end{cases} \]

The proposition below guarantees the existence of sets of Taylor–Wiles primes suitable for patching, generalizing [Reference Böckle, Harris, Khare and ThorneBHKT19, Proposition 5.19]. Actually there are some typos in the statement and proof of [Reference Böckle, Harris, Khare and ThorneBHKT19, Proposition 5.19]; the argument below can be used to rectify them.

Proof. If $S \cup Q = \emptyset$ then (i) and (ii) are vacuous and (iii) is an immediate consequence of Proposition 3.6(ii). For the rest of the proof, we assume that $S \cup Q$ is non-empty.

We apply [Reference ČesnaviciusČes15, Theorem 6.2], using that $H^0(\Gamma _{K, S \cup Q}, {\rm ad} ({\overline {\sigma }})(1)) = H^0(\Gamma _{K, S \cup Q}, {\rm ad} ({\overline {\sigma }})) = 0$ by the abundance assumption, and the local and global Euler characteristic formulas (note our assumption that $S \cup Q$ is non-empty) to deduce that

(3.3)\begin{equation} h^1_{{\mathcal{D}}_Q}(\Gamma_{K, S \cup Q} ,{\rm ad}({\overline{\sigma}})) - h^1_{{\mathcal{D}}_Q^{\perp}}(\Gamma_{K, S \cup Q} , {\rm ad} ({\overline{\sigma}})^{\vee}(1)) = \sum_{v \in S} (\dim {\mathcal{L}}_v - h^0(\Gamma_v, {\rm ad} ({\overline{\sigma}})) ) + \sum_{v \in Q} r . \end{equation}

The assumption that ${\mathcal {D}}_v$ is balanced for all $v \in S$ implies that $\dim {\mathcal {L}}_v - h^0(G_{F_v}, {\rm ad} ({\overline {\sigma }})) = 0$ for all $v \in S$. We claim that once we can arrange (i) and (ii), then (iii) follows automatically. Indeed, if $h^1_{{\mathcal {D}}_Q^{\perp }}(\Gamma _{K, S \cup Q} , {\rm ad} ({\overline {\sigma }})^{\vee }(1))$ vanishes then (3.3) implies that

\[ h^1_{{\mathcal{D}}_Q}(\Gamma_{K, S \cup Q} , {\rm ad} ({\overline{\sigma}})) = r \# Q = h^1_{{\mathcal{D}}}(\Gamma_{K, S}, {\rm ad}({\overline{\sigma}})) + (r-1)\# Q, \]

from which (iii) follows by invoking Proposition 3.6.

It therefore remains to show that $Q$ can be chosen so that

\begin{align*} &H^1_{{\mathcal{D}}_Q^{\perp}}(\Gamma_{K, S \cup Q} , {\rm ad}({\overline{\sigma}})^{\vee}(1))\\ &\quad :=\ker \bigg( H^1(\Gamma_{K, S \cup Q} , {\rm ad}({\overline{\sigma}})^{\vee}(1)) \rightarrow \bigg( \bigoplus_{v \in S} \frac{H^1(K_v, {\rm ad}({\overline{\sigma}})^{\vee}(1)) }{ L_v} \oplus \bigoplus_{v \in Q} H^1(K_v, {\rm ad}({\overline{\sigma}})^{\vee}(1)) \bigg) \bigg) \end{align*}

vanishes. By a comparison of inflation-restriction exact sequences, the inflation map $H^1(\Gamma _{K, S }, {\rm ad}({\overline {\sigma }})^{\vee }(1)) \rightarrow H^1(\Gamma _{K, S \cup Q}, {\rm ad}({\overline {\sigma }})^{\vee }(1))$ takes

\begin{align*} &\ker \bigg( H^1(\Gamma_{K, S} , {\rm ad}({\overline{\sigma}})^{\vee}(1)) \rightarrow \bigoplus_{v \in Q} H^1(k(v), {\rm ad}({\overline{\sigma}})^{\vee}(1)) \bigg)\\ &\quad \xrightarrow{\sim}\ker \bigg( H^1(\Gamma_{K, S \cup Q} , {\rm ad}({\overline{\sigma}})^{\vee}(1)) \rightarrow \bigoplus_{v \in Q} H^1(K_v, {\rm ad}({\overline{\sigma}})^{\vee}(1)) \bigg) . \end{align*}

Therefore it suffices by induction to show that for any $j$ and any non-zero $[\psi ] \in H^1(\Gamma _{K, S}, \widehat {\mathfrak {g}}_k^{\vee }(1))$, we can find infinitely many places $v \not \in S$ of $K$ such that $q_v \equiv 1 \text { mod }\ell ^j$, $\overline {\rho }(\operatorname {Frob}_v)$ is regular semisimple with connected centralizer in ${\widehat {G}}_k$, and $\operatorname {res}_{K_v} [\psi ] \neq 0$. The rest of the argument concludes exactly as in [Reference Böckle, Harris, Khare and ThorneBHKT19, Proof of Proposition 5.19].

4.1 Spaces of automorphic forms

We define integral spaces of automorphic forms as in [Reference Böckle, Harris, Khare and ThorneBHKT19, § 8.1]. For any open subgroup $U \subset G(\widehat {{\mathcal {O}}})$ and any $\mathbb {Z}[1/p]$-algebra $R$, we define:

• • $C(U, R)$ to be the $R$-module of functions $f : G(K) \backslash G({\mathbb {A}}_K) / U \to R$;

• • $C_c(U, R) \subset C(U, R)$ to be the $R$-submodule of functions $f$ which have finite support; and

• • $C_{\mathrm {cusp}}(U, R)$ to be the $R$-submodule of functions $f$ which are cuspidal, in the sense that for all proper parabolic subgroups $P \subset G$ and for all $g \in G({\mathbb {A}}_K)$, the integral

  \[ \int_{n \in N(K) \backslash N({\mathbb{A}}_K)} f(ng)\, dn \]
  vanishes, where $N$ is the unipotent radical of $P$.

This last integral is normalized by endowing $N(K) \backslash N({\mathbb {A}}_K)$ with its probability Haar measure (which makes sense because we are assuming that $p$ is a unit in $R$). We have $C_{\operatorname {cusp}}(U,R) \subset C_{c}(U, R)$ by [Reference Böckle, Harris, Khare and ThorneBHKT19, Proposition 8.2].

We define $C_{\operatorname {cusp}}(G,R) := \varinjlim _U C_{\mathrm {cusp}}(U, R)$.

Let $N = \sum _v n_v \cdot v \subset X$ be an effective divisor and let $U(N) := \ker ( \prod _v G({\mathcal {O}}_{K_v}) \to G({\mathcal {O}}_N) )$. The underlying set of places $|N| = \bigcup _{n_v>0} \{v\}$ will play the role of the set $S$ from § 3.

4.2 The excursion algebra and Lafforgue's parametrization

For the remainder of this section we choose a prime $\ell \neq p$ and coefficient field $E \subset \overline {{\mathbb {Q}}}_\ell$. For $R \in \{k, {\mathcal {O}},E\}$ we denote by ${\mathcal {B}}(U(N), R)$ the $R$-subalgebra of $\operatorname {End}_{R}(C_{\mathrm {cusp}}(U(N), R))$ generated by V. Lafforgue's excursion operators, as in [Reference Böckle, Harris, Khare and ThorneBHKT19, § 8.4]. For any fixed $N$, this is a finite $R$-algebra. The Hecke operators $T_{V,v}$ as lying in ${\mathcal {B}}(U(N), R)$, for $v \notin |N|$ and $V$ a representation of ${\widehat {G}}$, are defined as in [Reference Böckle, Harris, Khare and ThorneBHKT19].

The points of $\operatorname {Spec} {\mathcal {B}}(U(N),R)$ are naturally identified with semisimple $L$-parameters, in a manner that we will presently review. Combining [Reference Böckle, Harris, Khare and ThorneBHKT19, Corollaries 8.6, 8.11], we have the following (with notation as in those corollaries).

For $R \in \{k, E \}$ and any $N$, V. Lafforgue constructs a decomposition of $C_{\operatorname {cusp}}(U(N), \overline {R})$ into summands indexed by $L$-parameters $\sigma \in H^1(\Gamma _{K,N} , \widehat {G}(\overline {R}))$. More specifically, this decomposition comes from the generalized eigenspace decomposition for the action of ${\mathcal {B}}(U(N), R)$ on $C_{\operatorname {cusp}}(U(N), R)$. If $\Pi \subset C_{\operatorname {cusp}}(G, \overline {R})$ is a subspace such that $\Pi ^{U(N)}$ is stable under the ${\mathcal {B}}(U(N), \overline {R})$-action and supported over a unique maximal ideal, we will denote this maximal ideal by ${\mathfrak m} = {\mathfrak m}_\Pi \subset {\mathcal {B}}(U(N), \overline {R})$. The corresponding $L$-parameter is then denoted $\sigma _{\Pi } := \sigma _{{\mathfrak m}}$, and we say that $\sigma _{\Pi }$ is the $L$-parameter attached to $\Pi$. This assignment $\Pi \mapsto \sigma _{\Pi }$ is independent of the choice of $N$ such that $\Pi ^{U(N)} \neq 0$. This correspondence has the property that $\Pi$ and $\sigma _{\Pi }$ match under the local Langlands correspondence at all places where $\Pi$ is unramified, although we caution that this property does not characterize it for general groups.

4.3 Automorphy lifting theorems

Let $\chi \colon {\mathcal {B}}(U(N), {\mathcal {O}}) \rightarrow \overline {\mathbb {Z}}_{\ell }$ be a homomorphism. After possibly enlarging $E$, we can assume that $\chi$ takes values in ${\mathcal {O}}$. Let $\mathfrak {m} \subset {\mathcal {B}}(U(N), {\mathcal {O}})$ be the maximal ideal which is the kernel of the composition of $\chi$ with ${\mathcal {O}} \rightarrow k$, and ${\overline {\sigma }}_{\mathfrak {m}}$ the Galois representation corresponding to $\mathfrak {m}$ under Theorem 4.1. After possibly further enlarging $E$, we can assume that ${\overline {\sigma }}_{\mathfrak {m}}$ takes values in $\widehat {G}(k)$.

With Definition 3.7 in mind, we make the following assumptions on ${\overline {\sigma }}_{\mathfrak {m}}$:

1. (i) $\ell \nmid \# W$. This implies, in particular, that $\ell$ is a very good characteristic for ${\widehat {G}}$.

2. (ii) The subgroup $Z_{{\widehat {G}}^\text {ad}}(\overline {\sigma }_{\mathfrak m}(\Gamma _{K,|N|}))$ of ${\widehat {G}}_k$ is scheme-theoretically trivial.

3. (iii) The representation $\overline {\sigma }_{\mathfrak m}$ is absolutely ${\widehat {G}}$-irreducible.

4. (iv) The subgroup $\overline {\sigma }_{\mathfrak m}(\Gamma _{K(\zeta _\ell ), |N|})$ of ${\widehat {G}}(k)$ is ${\widehat {G}}$-abundant.

In addition, we choose a global deformation problem ${\mathcal {D}} := (|N|, \{{\mathcal {D}}_v\}_{v \in |N|})$ for ${\overline {\sigma }}_{\mathfrak {m}}$ such that all places of ramification for ${\overline {\sigma }}_{\mathfrak {m}}$ are contained in $|N|$, and:

1. (v) For each $v \in |N|$, the local deformation problem ${\mathcal {D}}_v$ is balanced in the sense of Definition 3.4 and unrestricted in the sense of Example 3.5. (So we require that $H^2(\Gamma _{K_v}, {\rm ad} \,\overline {\sigma }) = 0$ for $v \in |N|$.)

Let $R_{\overline {\sigma }_{\mathfrak m}, |N|, {\mathcal {D}}}$ be the corresponding global deformation ring (Proposition 3.6). From [Reference Böckle, Harris, Khare and ThorneBHKT19, Theorem 8.5] we have a pseudocharacter $\Theta _{U(N)} = (\Theta _{U(N),n})_{n \geq 1}$ valued in ${\mathcal {B}}(U(N), {\mathcal {O}})$, which factors through the quotient $\Gamma _K \to \Gamma _{K, |N|}$. Write $\Theta _{U(N), \mathfrak {m}}$ for the projection of the pseudocharacter $\Theta _{U(N)}$ to ${\mathcal {B}}(U(N), {\mathcal {O}})_{\mathfrak {m}}$, and $\sigma ^{\mathrm {univ}} \colon \Gamma _K \rightarrow \widehat {G}(R_{\overline {\sigma }_{\mathfrak m}, |N|, {\mathcal {D}}})$ for a representative of the universal deformation.

The following lemma is a variant of [Reference Böckle, Harris, Khare and ThorneBHKT19, Lemma 8.19].

We can now prove the following generalization of [Reference Böckle, Harris, Khare and ThorneBHKT19, Theorem 8.20] that allows for ramification.

Proof. In the case that $\overline {\sigma }_{\mathfrak m}$ is strongly ${\widehat {G}}$-irreducible, this is exactly [Reference Böckle, Harris, Khare and ThorneBHKT19, Theorem 8.17]. Without this assumption, [Reference Böckle, Harris, Khare and ThorneBHKT19, Lemma 8.18] shows that $C_c(W, {\mathcal {O}})$ is a free ${\mathcal {O}}[U/W]$-module (though of infinite rank). Let $u$ be a place of $K$ such that $W_u = G({\mathcal {O}}_{K_u})$, and let ${\mathcal {H}}_{G, u}$ denote the unramified Hecke algebra at $u$ with ${\mathcal {O}}$-coefficients. Then $C_c(W, {\mathcal {O}})$ is a finite ${\mathcal {H}}_{G, u}$-module [Reference XueXue20b, Theorem 0.0.3]. According to [Reference XueXue20b, § 8], it is possible to define an ${\mathcal {O}}$-subalgebra ${\mathcal {B}}_c(W, {\mathcal {O}}) \leq \operatorname {End}_{\mathcal {O}}(C_c(W, {\mathcal {O}}))$ of excursion operators, leaving invariant $C_{\mathrm {cusp}}(W, {\mathcal {O}})$, and extending ${\mathcal {B}}(W, {\mathcal {O}})$. Moreover, this ring contains the image of ${\mathcal {H}}_{G, u}$ in $\operatorname {End}_{\mathcal {O}}(C_c(W, {\mathcal {O}}))$, showing that ${\mathcal {B}}_c(W, {\mathcal {O}})$ is a finite ${\mathcal {H}}_{G, u}$-algebra and that $C_c(W, {\mathcal {O}})$ is a finite ${\mathcal {B}}_c(W, {\mathcal {O}})$-module.

Let us identify ${\mathfrak m}$ with its pre-image in ${\mathcal {B}}_c(W, {\mathcal {O}})$ under the canonical surjection ${\mathcal {B}}_c(W, {\mathcal {O}}) \to {\mathcal {B}}(W, {\mathcal {O}})$ (given by restricting excursion operators to $C_{\mathrm {cusp}}(W, {\mathcal {O}})$). To prove the lemma, it is enough to show that $C_{\mathrm {cusp}}(W, {\mathcal {O}})_{\mathfrak m}$ is a direct summand ${\mathcal {B}}_c(W, {\mathcal {O}})$-submodule of the free ${\mathcal {O}}[U / W]$-module $C_c(W, {\mathcal {O}})$. Using the constructions of [Reference XueXue20b], we see that there is a short exact sequence, respecting the action of excursion operators:

\[ 0 \to C_{\mathrm{cusp}}(W, {\mathcal{O}}) \to C_c(W, {\mathcal{O}}) \overset{\prod_{P \leq G} c_P}{\to} \prod_{P \leq G} C(U(\mathbb{A}_K) M(K) \backslash G(\mathbb{A}_K / W, {\mathcal{O}}), \]

where the sum ranges over a set of representatives for the conjugacy classes of proper parabolic subgroups $P = M U$ of $G$, and $c_P$ denotes the corresponding constant term morphism. Let $\mathscr {C}$ denote the image of $C_c(W, {\mathcal {O}})$ under $\prod _{P \leq G} c_P$, a ${\mathcal {B}}_c(W, {\mathcal {O}})$-module. The proof will be complete if we can show that $\mathscr {C}_{\mathfrak m} = 0$. This follows from the absolute ${\widehat {G}}$-irreducibility of $\overline {\sigma }_{\mathfrak m}$. Indeed, if $\mathscr {C}_{\mathfrak m} \neq 0$, then the compatibility of excursion operators with constant term morphisms ([Reference XueXue20b, § 8.4]) shows, along exactly the same lines as in [Reference XueXue20a, § 4.2], that $\overline {\sigma }_{\mathfrak m}$ must factor through $\hat {M}(\overline {\mathbb {F}}_\ell )$ for some proper parabolic $P = M U$ of $G$. This is a contradiction to the absolute ${\widehat {G}}$-irreducibility of $\overline {\sigma }_{\mathfrak m}$.

We begin by preparing the usual setup for patching. Recall that we have fixed a choice $T \subset B \subset G$ of split maximal torus and Borel subgroup of $G$. For a Taylor–Wiles datum $(Q, \{\varphi _v\}_{v \in Q})$, as in § 3.5, with $Q$ disjoint from $|N|$, we introduce the following notation. Extend the global deformation problem ${\mathcal {D}} = (|N|, \{{\mathcal {D}}_v\}_{v \in |N|})$ to ${\mathcal {D}}_Q := (|N| \cup Q, \{{\mathcal {D}}_{Q,v}\}_{v\in |N| \cup Q})$ by setting ${\mathcal {D}}_{Q, v} = {\mathcal {D}}_v$ if $v \in |N|$ and ${\mathcal {D}}_{Q,v} = {\mathcal {D}}_{\overline {\sigma },v}^{\square }$ (the unrestricted deformation problem) if $v \in Q$. Define $\Delta _Q$ to be the maximal $\ell$-power order quotient of the group $\prod _{v \in Q} T(k(v))$. Using local class field theory, the action of the universal deformation ring for the restriction of $\overline {\sigma }_{\mathfrak m}$ to the tame inertia groups at places in $Q$ equips $R_{\overline {\sigma }_{\mathfrak m}, |N| \cup Q, {\mathcal {D}}_Q}$ with a ${\mathcal {O}}[\Delta _Q]$-algebra structure. Writing ${\mathfrak a}_Q \subset {\mathcal {O}}[\Delta _Q]$ for the augmentation ideal, we have a canonical isomorphism $R_{\overline {\sigma }_{\mathfrak m}, |N| \cup Q, {\mathcal {D}}_Q} / {\mathfrak a}_Q \cong R_{\overline {\sigma }_{\mathfrak m}, |N|, {\mathcal {D}}}$.

We now define the relevant level structures for patching. We define open compact subgroups $U_1(Q) \subset U_0(Q) \subset U(N)$ as follows:

• • $U_0(Q) = \prod _v U_0(Q)_v$, where $U_0(Q)_v = U(N)_v = \ker ( G({\mathcal {O}}_{K_v}) \rightarrow G({\mathcal {O}}_{n_v \cdot v}))$ if $v \not \in Q$, and $U_0(Q)_v$ is an Iwahori group if $v \in Q$;

• • $U_1(Q) = \prod _v U_1(Q)_v$, where $U_1(Q)_v = U(N)_v$ if $v \not \in Q$, and $U_1(Q)_v$ is the maximal pro-prime-to-$\ell$ subgroup of $U_0(Q)_v$ if $v \in Q$.

Thus $U_1(Q) \triangleleft U_0(Q)$ is a normal subgroup, and there is a canonical isomorphism $U_0 / U_1 \cong \Delta _Q$.

We fix a place $v_0$ of $K$, and write $\ell ^M$ for the order of the $\ell$-Sylow subgroup of $G(\mathbb {F}_{q_{v_0}})$.

We now need to define auxiliary spaces of modular forms. We define $H'_0 = C_{\mathrm {cusp}}(U(N), {\mathcal {O}})_{\mathfrak m}$. There are surjective maps ${\mathcal {B}}(U_1(Q), {\mathcal {O}}) \twoheadrightarrow {\mathcal {B}}(U_0(Q), {\mathcal {O}}) \twoheadrightarrow {\mathcal {B}}(U(N), {\mathcal {O}})$, and we write ${\mathfrak m}$ as well for the pullback of ${\mathfrak m} \subset {\mathcal {B}}(U(N), {\mathcal {O}})$ to these two algebras. As in Lemma 4.5, we have surjective morphisms

\[ R_{\overline{\sigma}_{\mathfrak m}, |N| \cup Q, {\mathcal{D}}_Q} \twoheadrightarrow {\mathcal{B}}(U_1(Q), {\mathcal{O}})_{\mathfrak m} \twoheadrightarrow {\mathcal{B}}(U_0(Q), {\mathcal{O}})_{\mathfrak m}. \]

We define $H'_{Q, 1} := C_{\mathrm {cusp}}(U_1(Q), {\mathcal {O}})_{\mathfrak m}$ and $H'_{Q, 0} := C_{\mathrm {cusp}}(U_0(Q), {\mathcal {O}})_{\mathfrak m}$.

We discuss how to cut $H'_0$ out of $H'_{Q,0}$ as a direct summand, and an analogous construction for $H'_{Q,1}$. This will make use of the Hecke algebras ${\mathcal {H}}_{U_0(Q),v}$ at a place $v \in Q$ (where it is the Iwahori–Hecke algebra by our choice of level structure) and ${\mathcal {H}}_{U_1(Q),v}$. The ‘translation part’ of Bernstein's presentation of the Iwahori–Hecke algebra is an embedding ${\mathcal {O}}[X_\ast (T)] \to {\mathcal {H}}_{U_0(Q)_v}$, whose action on $H'_{Q,0}$ for $v \in Q$ induces an $R_{\overline {\sigma }_{\mathfrak m}, |N| \cup Q, {\mathcal {D}}_Q}[ \prod _{v \in Q} X_\ast (T) ]$-module structure on $H'_{Q, 0}$. We write ${\mathfrak n}_{Q, 0} \subset {\mathcal {O}}[ \prod _{v \in Q} X_\ast (T) ]$ for the maximal ideal which is associated to the tuple of characters ($v \in Q$):

(4.2)\begin{equation} \varphi_v^{-1} \circ \overline{\sigma}_{\mathfrak m}|_{W_{K_v}} : W_{K_v} \to {\widehat{T}}(k). \end{equation}

Then $H'_{Q, 0, {\mathfrak n}_{Q, 0}}$ is a direct factor $R_{\overline {\sigma }_{\mathfrak m}, |N| \cup Q, {\mathcal {D}}_Q}$-module of $H'_{Q, 0}$, and there is a canonical isomorphism $H'_{Q, 0, {\mathfrak n}_{Q, 0}} \cong H'_0$ of $R_{\overline {\sigma }_{\mathfrak m}, |N| \cup Q, {\mathcal {D}}_Q}$-modules.Footnote ²

Similarly, for $v \in Q$ we write $T(K_v)_\ell$ for the quotient of $T(K_v)$ by its maximal pro-prime-to-$\ell$ subgroup. Then there is a structure of $R_{\overline {\sigma }_{\mathfrak m}, |N| \cup Q, {\mathcal {D}}_Q}[ \prod _{v \in Q} T(K_v)_\ell ]$-module on $H'_{Q, 1}$, where the copy of $T(K_v)_\ell$ corresponding to $v \in Q$ acts via the analogous embedding ${\mathcal {O}}[T(K_v)_\ell ] \to {\mathcal {H}}_{U_1(Q)_v}$. We write ${\mathfrak n}_{Q, 1} \subset {\mathcal {O}}[ \prod _{v \in Q} T(K_v)_\ell ]$ for the maximal ideal which is associated to the tuple of characters (4.2). Then $H'_{Q, 1, {\mathfrak n}_{Q, 1}}$ is a direct factor $R_{\overline {\sigma }_{\mathfrak m}, |N| \cup Q, {\mathcal {D}}_Q}$-module of $H'_{Q, 1}$, and the two structures of ${\mathcal {O}}[\Delta _Q]$-module on $H'_{Q, 1, {\mathfrak n}_{Q, 1}}$, one arising from the homomorphism ${\mathcal {O}}[\Delta _Q] \to R_{\overline {\sigma }_{\mathfrak m}, |N| \cup Q, {\mathcal {D}}_Q}$ and the other from the homomorphism ${\mathcal {O}}[\Delta _Q] \to {\mathcal {O}}[\prod _{v \in Q} T(K_v)_\ell ]$, are the same.

We need two key properties of the modules $H'_{Q, 1}$ (with fixed $j$ as in Proposition 3.9):

• • The natural inclusion

  \[ C_{\mathrm{cusp}}(U_0(Q), {\mathcal{O}}) \subset C_{\mathrm{cusp}}(U_1(Q), {\mathcal{O}}) \]
  induces an identification $H'_{Q, 0, {\mathfrak n}_{Q, 0}} = (H'_{Q, 1, {\mathfrak n}_{Q, 1}})^{\Delta _Q}$.

• • By Lemma 4.7, we then have that if $q_v \equiv 1 \text { mod }\ell ^{M'}$ for each $v \in Q$ and for some $M' \geq M$, then $(H'_{Q, 1})^{\ell ^{j} \Delta _Q}$ is a free ${\mathcal {O}}[\Delta _Q / \ell ^{j} \Delta _Q]$-module for each $0 \leq j \leq M' - M$. This property implies in turn that $(H'_{Q, 1, {\mathfrak n}_{Q, 1}})^{\ell ^{j} \Delta _Q}$ is a free ${\mathcal {O}}[\Delta _Q / \ell ^{j} \Delta _Q]$-module. Observe that $\Delta _Q / \ell ^{j} \Delta _Q \cong ({\mathbb {Z}} / \ell ^{j} {\mathbb {Z}})^{\oplus r \# Q}$, where $r = \operatorname {rank} {\widehat {G}}$.

For patching it is a bit more convenient to work with the modules $H_Q := \operatorname {Hom}_{\mathcal {O}}((H'_{Q, 1, {\mathfrak n}_{Q, 1}})^{\ell ^{j} \Delta _Q}, {\mathcal {O}})$ and $H_0 := \operatorname {Hom}_{\mathcal {O}}(H'_0, {\mathcal {O}})$. These are finite free ${\mathcal {O}}$-modules, which we endow with their natural structures of $R_{\overline {\sigma }_{\mathfrak m}, |N| \cup Q, {\mathcal {D}}_Q} \otimes _{{\mathcal {O}}[\Delta _Q]} {\mathcal {O}}[\Delta _Q / \ell ^{j} \Delta _Q]$-module and $R_{\overline {\sigma }_{\mathfrak m}, |N|, {\mathcal {D}}_Q}$-module, respectively, by dualization. We can summarize the preceding discussion as follows:

• • The module $H_Q$ is a finite free ${\mathcal {O}}[\Delta _Q / \ell ^{j} \Delta _Q]$-module, where ${\mathcal {O}}[\Delta _Q / \ell ^{j} \Delta _Q]$ acts via the algebra homomorphism

  \[ {\mathcal{O}}[\Delta_Q / \ell^{j} \Delta_Q] \to R_{\overline{\sigma}_{\mathfrak m}, |N| \cup Q, {\mathcal{D}}_Q} \otimes_{{\mathcal{O}}[\Delta_Q]} {\mathcal{O}}[\Delta_Q / \ell^{j} \Delta_Q]. \]

• • There is a natural surjective map $H_Q \twoheadrightarrow H_0$, which factors through an isomorphism $(H_Q)_{\Delta _Q} \xrightarrow {\sim } H_0$, and is compatible with the isomorphism $R_{\overline {\sigma }_{\mathfrak m}, |N| \cup Q, {\mathcal {D}}_Q} / {\mathfrak a}_Q \cong R_{\overline {\sigma }_{\mathfrak m}, |N|, {\mathcal {D}}}$.

Let $h := h^1_{{\mathcal {D}}}(\Gamma _{K, |N|}, {\rm ad} \,{\overline {\sigma }})$. By Proposition 3.9, we can find for each $j \geq 1$ a Taylor–Wiles datum $(Q_j, \{ \varphi _v \}_{v \in Q_j})$ which satisfies the following conditions:

• • $Q_j$ is disjoint from $|N|$.

• • For each $v \in Q_N$, we have $q_v \equiv 1 \text { mod }\ell ^{j + M}$ and $\# Q_j= h$.

• • There exists a surjection ${\mathcal {O}}[[x_1, \ldots, x_g]]\twoheadrightarrow R_{\overline {\sigma }_{\mathfrak m}, |N| \cup Q_j, {\mathcal {D}}}$, where $g = hr$.

Define $R_\infty = {\mathcal {O}} \unicode{x27E6} X_1, \ldots, X_g \unicode{x27E7}$. The situation is summarized in the following diagram.

We now patch these objects together. This involves quotienting the objects in the diagram by open ideals to get diagrams of Artinian objects. Then because there are only finitely many isomorphism classes of such diagrams, one can pass to the inverse limit. The details of this process are the same as in [Reference Böckle, Harris, Khare and ThorneBHKT19, p. 48–49], so we skip to the conclusion. Define $\Delta _\infty := {\mathbb {Z}}_\ell ^g$, $\Delta _j := \Delta _\infty / \ell ^{j}\Delta _\infty$, $S_\infty := {\mathcal {O}}\unicode{x27E6} \Delta _\infty \unicode{x27E7}$, ${\mathfrak b}_j := \ker (S_\infty \to {\mathcal {O}}[[\Delta _j]])$, and ${\mathfrak b}_0 := \ker ( S_\infty \to {\mathcal {O}})$. By patching, we have the following objects:

• • $R^\infty$, a complete Noetherian local ${\mathcal {O}}$-algebra with residue field $k$, which is equipped with structures of $S_\infty$-algebra and a surjective map $R_\infty \twoheadrightarrow R^\infty$;

• • $H_\infty$, a finite $R^\infty$-module;

• • $\alpha _\infty$, an isomorphism $R^\infty / {\mathfrak b}_0 \cong R_{\overline {\sigma }_{\mathfrak m}, |N|, {\mathcal {D}}}$;

• • $\beta _\infty$, an isomorphism $H_\infty / {\mathfrak b}_0 \cong H_0$.

These objects have the following additional properties:

• • $H_\infty$ is free as an $S_\infty$-module.

• • The isomorphisms $\alpha _\infty$, $\beta _\infty$ are compatible with the structure of $R_{\overline {\sigma }_{\mathfrak m}, |N|, {\mathcal {D}}}$-module on $H_0$.

The situation is summarized in the following diagram.

Since $S_{\infty }$ is topologically free, we may choose the dashed arrow so that the leftmost triangle commutes.

We find that

\[ \dim R^\infty \geq \operatorname{depth}_{R^\infty} H_\infty \geq \operatorname{depth}_{S_\infty} H_\infty = \dim S_\infty = \dim R_\infty \geq \dim R^\infty, \]

and hence that these inequalities are equalities, $R_\infty \to R^\infty$ is an isomorphism, and (by the Auslander–Buchsbaum formula) $H_\infty$ is also a free $R^\infty$-module. It follows that $H_\infty / {\mathfrak b}_0 \cong H_0$ is a free $R^\infty / {\mathfrak b}_0 \cong R_{\overline {\sigma }_{\mathfrak m}, |N|, {\mathcal {D}}}$-module, and that $R_{\overline {\sigma }_{\mathfrak m}, |N|, {\mathcal {D}}}$ is an ${\mathcal {O}}$-flat complete intersection. This in turn implies that $C_{\mathrm {cusp}}(U, {\mathcal {O}})_{\mathfrak m} \cong \operatorname {Hom}_{\mathcal {O}}(H_0, {\mathcal {O}})$ is a free $R_{\overline {\sigma }_{\mathfrak m}, |N|, {\mathcal {D}}}$-module (since complete intersections are Gorenstein). This completes the proof of the theorem.

5.1 Approach based on lifting theorems for Galois representations

First we show that Theorem 5.1 follows from the same statement when $G = \operatorname {GL}_n$. Pick a faithful irreducible representation of $G$ on a finite projective $\mathbb {Z}$-module, say of rank $n > 1$. This induces an injection $\mathfrak {g} \hookrightarrow \mathfrak {gl}_n$, which is split over $\mathbb {Z}_{\ell }$ for all sufficiently large $\ell$ and therefore induces an injection $H^2({\Gamma _{K_v}}, {\rm ad}({\overline {\sigma }}_{\ell })) \hookrightarrow H^2({\Gamma _{K_v}}, \mathfrak {gl}_n \circ {\overline {\sigma }}_{\ell })$. This reduces the claim to the case $\widehat {G} = \operatorname {GL}_n$, so for the rest of this subsection we focus on the case $G = \operatorname {GL}_n$.

5.1.1 Some linear algebra

We recall the following result of Deligne [Reference DeligneDel80, Proposition 1.6.1].

Proposition 5.2 Let $k$ be a field and let $V$ be a finite-dimensional $k$-vector space, equipped with a nilpotent endomorphism $N : V \to V$. Then there exists a unique increasing filtration $M_\bullet$ of $V$ such that $N M_i \subset M_{i-2}$ and, for each $k \geq 0$, $N^k$ induces isomorphisms $\operatorname {gr}_k M_\bullet \to \operatorname {gr}_{-k} M_\bullet$.

If $N$ is a nilpotent endomorphism of an $n$-dimensional vector space, we will write $\operatorname {Jord}(N)$ for the partition of $n$ given by the sizes of the Jordan blocks of $N$. We recall that there is a partial order on partitions $n$ which can be viewed as corresponding to the closure ordering of nilpotent orbits in the adjoint representation of $\operatorname {GL}_n$.

Let $v$ be a place of $K$, and suppose given a prime $\ell > n$ not dividing $q$. Let $E / {\mathbb {Q}}_\ell$ be a finite extension with residue field $k_E = {\mathcal {O}}_E / (\varpi _E)$, and suppose given a continuous representation $\rho : G_{K_v} \to \operatorname {GL}_n({\mathcal {O}}_E)$. Fix a choice of (geometric) Frobenius lift $\phi _v \in G_{K_v}$, and $t_\ell \in G_{K_v}$ a generator for the $\ell$-part of tame inertia. Let us say that $\rho$ is good if the following conditions are satisfied:

• • $\rho$ is unipotently ramified. Let $N = \log (\rho (t_\ell )) \in M_n({\mathcal {O}}_E)$; then $\phi _v N \phi _v^{-1} = q^{-1} N$.

• • The eigenvalues of $\rho (\phi _v)$ all lie in $E$.

• • $\rho$ is pure of weight 0. This means that the eigenvalues of $\rho (\phi _v)$ on $V = E^n$ are $q_v$-Weil numbers and that, writing $V_i \subset V$ for the subspace where the eigenvalues are of weight $i$, $N^k$ induces an isomorphism $V_k \to V_{-k}$ for each $k \geq 0$. (Note that the filtration associated to $N$ by Proposition 5.2 is then $M_k = \bigoplus _{i \leq k} V_i$.)

• • Let $f_i(X) = \det (X - \rho (\phi _v)|_{V_i}) \in {\mathcal {O}}_E[X]$. Then for each $i \neq j$, the reduced polynomials $\overline {f}_i(X), \overline {f}_j(X) \in k_E[X]$ are coprime.

• • For each $i, j$, the polynomials $\overline {f}_i(X)$, $\overline {f}_j(q_v^{-1} X) \in k_E[X]$ fail to be coprime only if $j = i + 2$.

Let $L = {\mathcal {O}}_E^n$, so that $V = L \otimes _{{\mathcal {O}}_E} E$, and define $L_i = V_i \cap L$. Hensel's lemma implies that if $\rho$ is good then $L = \bigoplus _{i \in {\mathbb {Z}}} L_i$.

Lemma 5.3 Suppose that $\rho$ is good, and let $\overline {N} \in M_n(k_E)$ denote the reduction of $N$ modulo $\varpi _E$. If $\operatorname {Jord}(\overline {N}) = \operatorname {Jord}(N)$, then $H^2(K_v, {\rm ad} \,\overline {\rho }) = 0$.

Proof. The equality $\operatorname {Jord}(\overline {N}) = \operatorname {Jord}(N)$ is equivalent to the equalities $\dim _E \ker N^k = \dim _{k_E} \ker \overline {N}^k$ for each $k \geq 0$. Let $\overline {L} = L \otimes _{{\mathcal {O}}_E} k_E$ and $\overline {L}_i = L_i \otimes _{{\mathcal {O}}_E} k_E$. We first claim that for each $k \geq 0$, $\overline {N}^k$ induces an isomorphism $\overline {L}_k \to \overline {L}_{-k}$. Equivalently, the intersection $(\ker \overline {N}^k) \cap \overline {L}_k$ is 0.

The equality $\dim _E \ker N^k = \dim _{k_E} \ker \overline {N}^k$ implies that in fact $\ker \overline {N}^k = (\ker N^k \cap L) \otimes _{{\mathcal {O}}_E} k_E$. Since $\ker N^k$ is invariant under $\operatorname {Ad} \rho (\phi _v)$, we find that in fact

\[ (\ker \overline{N}^k) \cap \overline{L}_k = (\ker N^k \cap L_k) \otimes_{{\mathcal{O}}_E} k_E = 0. \]

This establishes the claim. By linear algebra, we then have a decomposition

\[ \overline{L} = \bigoplus_{i \in {\mathbb{Z}}} \bigoplus_{j=0}^i \overline{N}^j \overline{L}(i), \]

where $\overline {L}(i) = \overline {L}_i \cap (\ker \overline {N}^{i+1})$.

By Tate duality, we need to show that $H^0(K_v, {\rm ad} \,\overline {\rho }(1)) = \operatorname {Hom}_{G_{K_v}}(\overline {L}, \overline {L}(1)) = 0$. An element of this Hom space determines a linear map $F : \overline {L} \to \overline {L}$ which commutes with $\overline {N}$ and satisfies $F(\overline {L}_i) \subset \overline {L}_{i+2}$ for each $i \in {\mathbb {Z}}$. To show the Hom space vanishes, it is enough to show that $F$ annihilates each $\overline {L}(i)$. However, if $x \in \overline {L}(i)$ then we find $F(x) \in \overline {L}_{i+2}$ and

\[ \overline{N}^{i+1} F(x) = F( \overline{N}^{i+1} x ) = 0. \]

The restriction of $\overline {N}^{i+1}$ to $\overline {L}_{i+2}$ is injective, so this forces $F(x) = 0$, hence $F = 0$, as required.