Lattices in the cohomology of Shimura curves

Abstract

We prove the main conjectures of Breuil (J Reine Angew Math, 2012) (including a generalisation from the principal series to the cuspidal case) and Dembélé (J Reine Angew Math, 2012), subject to a mild global hypothesis that we make in order to apply certain $R=\mathbb {T}$ theorems. More precisely, we prove a multiplicity one result for the mod $p$ cohomology of a Shimura curve at Iwahori level, and we show that certain apparently globally defined lattices in the cohomology of Shimura curves are determined by the corresponding local $p$-adic Galois representations. We also indicate a new proof of the Buzzard–Diamond–Jarvis conjecture in generic cases. Our main tools are the geometric Breuil–Mézard philosophy developed in Emerton and Gee (J Inst Math Jussieu, 2012), and a new and more functorial perspective on the Taylor–Wiles–Kisin patching method. Along the way, we determine the tamely potentially Barsotti–Tate deformation rings of generic two-dimensional mod $p$ representations, generalising a result of Breuil and Mézard (Bull Soc Math de France, 2012) in the principal series case.

Appendices

Appendix A: Unipotent Fontaine–Laffaille modules and $\phi$-modules

1.1 Appendix A.1: Unipotent objects

Recall that results in $p$-adic Hodge theory that are valid in the Fontaine–Laffaille range can often be extended slightly to the so-called unipotent case (see below for specific examples of what we mean by this). In the proof of Lemma 7.4.2, we need to compare the Galois representations associated to certain unipotent $\varphi$-modules with the Galois representations associated to unipotent Fontaine–Laffaille modules. Breuil makes a similar comparison (without allowing the unipotent case) in the proofs of [15, Prop. 7.3] and [15, Prop. A.3], making use of certain results from [12]. In this appendix we will extend those results to the unipotent case, so that we can carry out the same argument. Since these extensions are minor, we will generally indicate where the proofs of [12] need to be changed, rather than repeating any proofs in their entirety. We attempt to keep our notation as consistent as possible with that of [12]; for instance we write $\phi$ in this appendix where we wrote $\varphi$ in the body of the paper.

Remark A.1.1

As in Appendix A of [15], there exist $\mathbb {F}$-coefficient versions of all of the following results (rather than just ${\mathbb {F}_p}$-coefficients); in fact we can take the coefficients to be any Artinian local ${\mathbb {F}_p}$-algebra (such as the ring $\mathbb {F}_J$ in the proof of Lemma 7.4.2). These more general results follow entirely formally from their ${\mathbb {F}_p}$-coefficient versions, and so to simplify our notation we will omit coefficients in what follows.

Throughout this appendix we let $k$ be a perfect field of characteristic $p > 0$, and let $K_0$ denote the fraction field of $W(k)$. Fix a uniformizer $\pi$ of $K_0$. We begin by recalling the definitions of several categories of objects that we wish to consider.

Definition A.1.2

Suppose that $0 \le h \le p-1$. We define the category $\underline{\mathrm {MF}}^{f,h}_k$ (the category of mod $p$ Fontaine–Laffaille modules of height $h$) to be the category whose objects consist of:

• a finite-dimensional $k$-vector space $M$,

• a filtration $(\mathop {\mathrm {Fil}}\nolimits ^i M)_{i \in \mathbb {Z}}$ such that $\mathop {\mathrm {Fil}}\nolimits ^i M = M$ for $i \le 0$ and $\mathop {\mathrm {Fil}}\nolimits ^i M = 0$ for $i \ge h+1$, and

• for each integer $i$, a semi-linear map $\phi _i : \mathop {\mathrm {Fil}}\nolimits ^i M \rightarrow M$ such that $\phi _i |_{\mathop {\mathrm {Fil}}\nolimits ^{i+1} M} = 0$ and $\sum \mathrm{Im}\,\phi _i = M$.

Morphisms in $\underline{\mathrm {MF}}^{f,h}_k$ are $k$-linear maps that are compatible with filtrations and commute with the $\phi _i$.

When $h \le p-2$, there is an exact and fully faithful functor from the category $\underline{\mathrm {MF}}^{f,h}_k$ to the category of ${\mathbb {F}_p}$-representations of the absolute Galois group of $K_0$; this is no longer the case when $h=p-1$, but as we will see, it remains true if we restrict to a certain subcategory of $\underline{\mathrm {MF}}^{f,p-1}_k$.

Definition A.1.3

We say that an object $M$ of $\underline{\mathrm {MF}}^{f,h}_k$ is multiplicative if $\mathop {\mathrm {Fil}}\nolimits ^{h} M = M$; we say that $M$ is unipotent if it has no nontrivial multiplicative quotients. Let $\underline{\mathrm {MF}}^{u,p-1}_k$ denote the full subcategory of $\underline{\mathrm {MF}}^{f,p-1}_k$ consisting of the unipotent objects.

We turn next to strongly divisible modules. Let $\overline{S}= k\langle u \rangle$ be the divided power polynomial algebra in the variable $u$, and let $\mathop {\mathrm {Fil}}\nolimits ^h \overline{S}$ be the ideal generated by $\gamma _i(u)$ for $i \ge h$. Write $c$ for the element $-\gamma _p(u)-(\pi /p) \in \overline{S}^{\times }$, and for $0 \le h \le p-1$ define $\phi _h : \mathop {\mathrm {Fil}}\nolimits ^h \overline{S}\rightarrow \overline{S}$ by setting $\phi _h(\gamma _h(u)) = c^h / h!$ and $\phi _h(\gamma _i(u)) = 0$ for $i > h$, except that when $h=p-1$ we set $\phi _{p-1}(\gamma _p(u)) = c^p /(p-1)! \in \overline{S}^{\times }$. Set $\phi = \phi _0$.

Definition A.1.4

Suppose that $0 \le h \le p-1$. We define the category $\underline{\fancyscript{M}}_{0,k}^h$ (the category of mod $p$ strongly divisible modules of height $h$) to be the category whose objects consist of:

• a free, finite rank $\overline{S}$-module $\fancyscript{M}$,

• a sub-$\overline{S}$-module $\mathop {\mathrm {Fil}}\nolimits ^h \fancyscript{M}$ of $\fancyscript{M}$ containing $(\mathop {\mathrm {Fil}}\nolimits ^h \overline{S})\fancyscript{M}$

• a $\phi$-semi-linear map $\phi _h : \mathop {\mathrm {Fil}}\nolimits ^h \fancyscript{M}\rightarrow \fancyscript{M}$ such that for each $s \in \mathop {\mathrm {Fil}}\nolimits ^h \overline{S}$ and $x \in \fancyscript{M}$ we have $\phi _h(sx) = c^{-h} \phi _h(s) \phi _h(u^h x)$ and such that $\phi _h(\mathop {\mathrm {Fil}}\nolimits ^h \fancyscript{M})$ generates $\fancyscript{M}$ over $\overline{S}$.

Morphisms in $\underline{\fancyscript{M}}_{0,k}^h$ are $\overline{S}$-linear maps that are compatible with filtrations and commute with $\phi _h$.

Definition A.1.5

An object of $\underline{\fancyscript{M}}_{0,k}^h$ is called multiplicative if $\mathop {\mathrm {Fil}}\nolimits ^h \fancyscript{M}= \fancyscript{M}$, and unipotent if it has no non-zero multiplicative quotients. Let $\underline{\fancyscript{M}}_{0,k}^{u,p-1}$ denote the full subcategory of $\underline{\fancyscript{M}}_{0,k}^{p-1}$ consisting of the unipotent objects.

Finally we define several categories of étale $\phi$ -modules.

Definition A.1.6

Let $\underline{\mathfrak {D}}$ be the category whose objects are finite-dimensional $k(\!(u)\!)$-vector spaces $\mathfrak {D}$ equipped with an injective map $\mathfrak {D} \rightarrow \mathfrak {D}$ that is semilinear with respect to the $p$th power map on $k(\!(u)\!)$; morphisms in $\underline{\mathfrak {D}}$ are $k(\!(u)\!)$-linear maps that commute with $\phi$.

Let $\underline{\mathfrak {M}}$ be the category whose objects are finite-dimensional $k [\![u ]\!]$-modules $\mathfrak {M}$ equipped with an injective map $\mathfrak {M} \rightarrow \mathfrak {M}$ that is semilinear with respect to the $p$th power map on $k [\![u ]\!]$; morphisms in $\underline{\mathfrak {M}}$ are $k [\![u ]\!]$-linear maps that commute with $\phi$.

Write $\phi ^*$ for the $k [\![u ]\!]$-linear map ${\mathrm{Id }}\otimes \phi : k [\![u ]\!]\otimes _{\phi } \mathfrak {M} \rightarrow \mathfrak {M}$. We say that an object $\mathfrak {M}$ of $\underline{\mathfrak {M}}$ has height $h$ if $\mathfrak {M}/\mathrm{Im}\,\!(\phi ^*)$ is killed by $u^h$. Write $\underline{\mathfrak {M}}^h$ for the full subcategory of $\underline{\mathfrak {M}}$ of objects of height $h$, and $\underline{\mathfrak {D}}^h$ for the full subcategory of $\underline{\mathfrak {D}}$ consisting of objects isomorphic to $\mathfrak {M}[\frac{1}{u}]$ for some $\mathfrak {M} \in \underline{\mathfrak {M}}^h$.

Definition A.1.7

We say that an object $\mathfrak {M}$ of $\underline{\mathfrak {M}}^h$ is unipotent if it has no non-zero quotients $\mathfrak {M}'$ such that $\phi (\mathfrak {M}') \subset u^h \mathfrak {M}'$. Let $\underline{\mathfrak {M}}^{u,p-1}$ denote the full subcategory of $\underline{\mathfrak {M}}^{p-1}$ consisting of the unipotent objects; let $\underline{\mathfrak {D}}^{u,p-1}$ denote the full subcategory of $\underline{\mathfrak {D}}^{p-1}$ consisting of objects isomorphic to $\mathfrak {M}[\frac{1}{u}]$ for $\mathfrak {M} \in \underline{\mathfrak {M}}^{u,p-1}$.

We recall the following fundamental result of Fontaine.

Theorem A.1.8

([29, B.1.7.1]) The functor $\mathfrak {M} \leadsto \mathfrak {M}[\frac{1}{u}]$ is an equivalence of categories from $\underline{\mathfrak {M}}^h$ to $\underline{\mathfrak {D}}^h$ for $0 \le h \le p-2$, as well as from $\underline{\mathfrak {M}}^{u,p-1}$ to $\underline{\mathfrak {D}}^{u,p-1}$.

1.2 Appendix A.2: Generic fibres

We fix an algebraic closure $\overline{K}_0$ of $K_0$, and write $G_{K_0} = \mathrm {Gal}(\overline{K}_0/K_0)$. We also fix a system of elements $\pi _n$ in $\overline{K}_0$, for $n \ge 0$, such that $\pi _0 = \pi$ and $\pi _n^p = \pi _{n-1}$ for $n \ge 1$. We write $K_{\infty } = \cup _n K_0(\pi _n)$, and $G_{\infty } = \mathrm {Gal}(\overline{K}_0/K_\infty )$.

For each of the categories in Definitions A.1.2–A.1.7 we will now define a “generic fibre” functor to $k$-representations of $G_{K_0}$ (for Fontaine–Laffaille modules) or $G_{\infty }$ (for strongly divisible modules and $\phi$-modules).

Let $A_{{\mathrm{cris }}}$ be the ring (of the same name) defined in [10, §3.1.1]. The ring $A_{{\mathrm{cris }}}$ has a filtration $\mathop {\mathrm {Fil}}\nolimits ^h A_{{\mathrm{cris }}}$ for $h \ge 0$ and an endomorphism $\phi$ with the property that $\phi (\mathop {\mathrm {Fil}}\nolimits ^h A_{{\mathrm{cris }}}) \subset p^h A_{{\mathrm{cris }}}$ for $0 \le h \le p-1$; for such $h$ we may define $\phi _h : \mathop {\mathrm {Fil}}\nolimits ^h A_{{\mathrm{cris }}} \rightarrow A_{{\mathrm{cris }}}$ to be $\phi /p^h$. By [10, Lem. 3.1.2.2] the ring $A_{{\mathrm{cris }}}/pA_{{\mathrm{cris }}}$ is the ring denoted $R^{DP}$ in [12]. We have an induced filtration on $A_{{\mathrm{cris }}}/pA_{{\mathrm{cris }}}$ as well as induced maps $\phi _h : \mathop {\mathrm {Fil}}\nolimits ^h (A_{{\mathrm{cris }}}/pA_{{\mathrm{cris }}}) \rightarrow A_{{\mathrm{cris }}}/pA_{{\mathrm{cris }}}$ for $0 \le h \le p-1$. The ring $A_{{\mathrm{cris }}}$ (and so also $A_{{\mathrm{cris }}}/pA_{{\mathrm{cris }}}$) has a natural action of $G_{K_0}$ that commutes with all of the above structures. We regard $A_{{\mathrm{cris }}}/p A_{{\mathrm{cris }}}$ as an $\overline{{S}}$-algebra by sending the divided power $\gamma _i(u)$ to $\gamma _i(\underline{\pi })$, where $\underline{\pi } \in A_{{\mathrm{cris }}}/p A_{{\mathrm{cris }}}$ is the element defined in [12, §3.2]; then the group $G_{K_{\infty }}$ acts trivially on the image of $\overline{{S}}$ in $A_{{\mathrm{cris }}}/p A_{{\mathrm{cris }}}$.

Definition A.2.1

We define the following functors.

• (1) If $M \in \underline{\mathrm {MF}}^{f,h}_k$ for $0 \le h \le p-2$ or $M \in \underline{\mathrm {MF}}^{u,p-1}_k$, we set

  $$\begin{aligned} T(M) = \mathop {\mathrm {Hom}}\nolimits _{\mathop {\mathrm {Fil}}\nolimits ^{\bullet },\phi _{\bullet }}(M,A_{{\mathrm{cris }}}/pA_{{\mathrm{cris }}}), \end{aligned}$$

  the $k$-linear morphisms that preserve filtrations and commute with the $\phi _i$’s. This is a $G_{K_0}$-representation via $g(f)(x) = g(f(x))$ for $g \in G_{K_0}$, $f \in T(M)$, and $x \in M$.

• (2) If $\fancyscript{M}\in \underline{\fancyscript{M}}_{0,k}^h$ for $0 \le h \le p-2$ or $M \in \underline{\fancyscript{M}}_{0,k}^{u,h}$ with $h=p-1$, we set

  $$\begin{aligned} T(\fancyscript{M}) =\mathop {\mathrm {Hom}}\nolimits _{\mathop {\mathrm {Fil}}\nolimits ^{h},\phi _{h}}(\fancyscript{M},A_{{\mathrm{cris }}}/pA_{{\mathrm{cris }}}), \end{aligned}$$

  the $\overline{S}$-linear morphisms that preserve filtrations $\mathop {\mathrm {Fil}}\nolimits ^h$ and commute with $\phi _h$. This is a $G_{\infty }$-representation via $g(f)(x) = g(f(x))$ for $g \in G_{\infty }$, $f \in T(\fancyscript{M})$, and $x \in \fancyscript{M}$.

• (3) If $\mathfrak {D} \in \underline{\mathfrak {D}}$, we set

  $$\begin{aligned} T(\mathfrak {D}) = \mathop {\mathrm {Hom}}\nolimits _{\phi }(\mathfrak {D},k(\!(u)\!)^{\mathrm{sep }}), \end{aligned}$$

  the $k[[u]]$-linear morphisms that commute with $\phi$. This is naturally a $G_{\infty }$-representation by the theory of corps des normes (which gives an isomorphism between $\mathrm {Gal}(k(\!(u)\!)^{\mathrm{sep }}/k(\!(u)\!))$ and $G_{\infty }$; see [46, 3.2.3]).

• (4) If $\mathfrak {M} \in \underline{\mathfrak {M}}^{h}$ for $0 \le h \le p-2$ or $\mathfrak {M} \in \underline{\mathfrak {M}}^{u,p-1}$, we set $T(\mathfrak {M}) := T(\mathfrak {M}[\frac{1}{u}])$.

We refer to $T(M)$ (respectively $T(\fancyscript{M})$, $T(\mathfrak {D})$, $T(\mathfrak {M})$) as the generic fibre of $M$ (respectively $\fancyscript{M}$, $\mathfrak {D}$, $\mathfrak {M}$).

Proposition A.2.2

We have the following.

• (1) If $M \in \underline{\mathrm {MF}}^{f,h}_k$ for $0 \le h \le p-2$ or $M \in \underline{\mathrm {MF}}^{u,p-1}_k$, then

  $$\begin{aligned} \dim _k T(M) = \dim _k M. \end{aligned}$$

• (2) If $\fancyscript{M}\in \underline{\fancyscript{M}}_{0,k}^h$ for $0 \le h \le p-2$ or $\fancyscript{M}\in \underline{\fancyscript{M}}_{0,k}^{u,p-1}$, then

  $$\begin{aligned} \dim _k T(\fancyscript{M}) = \mathrm{rank }_{\overline{S}} \fancyscript{M}. \end{aligned}$$

• (3) If $\mathfrak {D} \in \underline{\mathfrak {D}}$, then

  $$\begin{aligned} \dim _k T(\mathfrak {D}) = \dim _{k(\!(u)\!)} \mathfrak {D}. \end{aligned}$$

Proof

Parts (1) and (3) are classical: for instance, (1) is [28, Thm. 6.1], and (3) is [29, Prop. 1.2.6]. (In (3) the functor $T$ is even an anti-equivalence of categories.)

As for (2), the case $0 \le h \le p-2$ is [12, Lem. 3.2.1]. The proof for $\underline{\fancyscript{M}}_{0,k}^{u,p-1}$ is exactly the same, except that the reference to [45, 2.3.2.2] must be supplemented with a reference to [45, Lem. 2.3.3.1]. $\square$

1.3 Appendix A.3: Equivalences

Regard $k[\![u ]\!]$ naturally as a subring of $\overline{S}$. Following [12, §4], if $0 \le h \le p-1$ we define a functor $\Theta _h : \underline{\mathfrak {M}}^h \rightarrow \underline{\fancyscript{M}}_{0,k}^h$ as follows. If $\mathfrak {M}$ is an object of $\underline{\mathfrak {M}}^h$, we define $\Theta _h(\underline{\mathfrak {M}})$ to be the object $\fancyscript{M}$ constructed as follows:

• $\fancyscript{M}= \overline{S}\otimes _{\phi } \mathfrak {M}$,

• $\mathop {\mathrm {Fil}}\nolimits ^h \fancyscript{M}= \{ y \in \fancyscript{M}\, : \, ({\mathrm{Id }}\otimes \phi )(y) \in (\mathop {\mathrm {Fil}}\nolimits ^h \overline{S})\otimes _{\phi } \mathfrak {M}\}$,

• $\phi _h : \mathop {\mathrm {Fil}}\nolimits ^h \fancyscript{M}\rightarrow \fancyscript{M}$ is the composition of ${\mathrm{Id }}\otimes \phi : \mathop {\mathrm {Fil}}\nolimits ^h \fancyscript{M}\rightarrow (\mathop {\mathrm {Fil}}\nolimits ^h \overline{S})\otimes _{\phi } \mathfrak {M}$ with $\phi _h \otimes {\mathrm{Id }}: (\mathop {\mathrm {Fil}}\nolimits ^h \overline{S})\otimes _{\phi } \mathfrak {M} \rightarrow \overline{S}\otimes _{\phi } \mathfrak {M} \cong \fancyscript{M}$.

Theorem A.3.1

The functor $\Theta _h$ induces an equivalence of categories from $\underline{\mathfrak {M}}^h$ to $\underline{\fancyscript{M}}_{0,k}^h$ for $0 \le h \le p-2$, as well as an equivalence of categories from $\underline{\mathfrak {M}}^{u,p-1}$ to $\underline{\fancyscript{M}}_{0,k}^{u,p-1}$.

Proof

For $0 \le h \le p-2$ this is [12, Thm. 4.1.1], while the unipotent case is explained in the proof of Theorem 2.5.3 of [30]. (Note, however, that some of our terminology is dual to that of [30]: what we call a multiplicative object of $\underline{\fancyscript{M}}_{0,k}^h$ or $\underline{\mathrm {MF}}^{f,h}_k$, Gao calls étale.) We briefly recall the argument from [30]. If $A$ is the matrix of $\phi$ on $\mathfrak {M}$ with respect to some fixed basis, the condition that $\mathfrak {M}$ is unipotent is precisely the condition that the product $\prod _{n=1}^{\infty } \phi ^n (u^{p-1} A^{-1})$ converges to $0$, and this is exactly what is required for the proof of full faithfulness in [12, Thm. 4.1.1] to go through (as well as to show that $\Theta _h(\mathfrak {M})$ is actually unipotent); the essential subjectivity in [12, Thm. 4.1.1] is true when $h=p-1$ without any unipotence condition. $\square$

Proposition A.3.2

If $\mathfrak {M} \in \underline{\mathfrak {M}}^{u,p-1}$, we have a canonical isomorphism of $G_{\infty }$-representations $T(\mathfrak {M}) \cong T(\Theta _{p-1}(\mathfrak {M}))$.

Proof

The same statement for $\Theta _h : \underline{\mathfrak {M}}^h \rightarrow \underline{\fancyscript{M}}_{0,k}^h$ with $0 \le h \le p-2$ is [12, Prop. 4.2.1]. The proof in the unipotent case is identical, noting that the condition $\prod _{n=1}^{\infty } \phi ^n (u^{p-1} A^{-1})=0$ is precisely what is needed for the last two sentences of the argument in [12] to go through. $\square$

Following [12, §5], if $0 \le h \le p-1$ we define a functor $\fancyscript{F}_h : \underline{\mathrm {MF}}^{f,h}_k\rightarrow \underline{\fancyscript{M}}_{0,k}^h$ as follows. If $M$ is an object of $\underline{\mathrm {MF}}^{f,h}_k$, we define $\fancyscript{F}_h(M)$ to be the object $\fancyscript{M}$ constructed as follows:

• $\fancyscript{M}= \overline{S}\otimes _{k} M$,

• $\mathop {\mathrm {Fil}}\nolimits ^h \fancyscript{M}= \sum _{i=0}^h \mathop {\mathrm {Fil}}\nolimits ^i \overline{S}\otimes _k \mathop {\mathrm {Fil}}\nolimits ^{h-i} M$,

• $\phi _h = \sum _{i=0}^h \phi _i \otimes \phi _{h-i}$.

While we expect it to be true that if $M$ is unipotent then so is $\fancyscript{F}_{p-1}(M)$, and that the resulting functor $\fancyscript{F}^u_{p-1} : \underline{\mathrm {MF}}^{u,p-1}_k \rightarrow \underline{\fancyscript{M}}_{0,k}^{u,p-1}$ is fully faithful, we will not need these assertions, and so we do not prove them. Instead, we note the following.

Proposition A.3.3

If $M \in \underline{\mathrm {MF}}^{u,p-1}_k$ and $\fancyscript{F}_{p-1}(M)$ is unipotent, then we have a canonical isomorphism of $G_{\infty }$-representations $T(M)|_{G_{\infty }} \cong T(\fancyscript{F}_{p-1}(M))$.

Proof

The same argument as in the paragraph before Lemme 5.1 of [12] (i.e., the proof of [10, 3.2.1.1]) shows that the natural map $T(M) \rightarrow T(\fancyscript{F}_{p-1}(M))$ is injective. Now the result follows by parts (1) and (2) of Proposition A.2.2 together with the assumption that $\fancyscript{F}_{p-1}(M)$ is unipotent. $\square$

Appendix B: Remarks on the geometric Breuil–Mézard philosophy

In this appendix we will explain some variants on the arguments of the main part of the paper, which are either more conceptual, or which avoid the use of results from papers such as [4, 33, 34]. Since nothing in the main body of the paper depends on this appendix, and since a full presentation of the arguments would be rather long, we only sketch the proofs.

1.1 Appendix B.1: Avoiding the use of results on the weight part of Serre’s conjecture

In this section we will indicate how the main results of the paper (namely, the application of the results of Sects. 8–10 to the specific examples of patching functors constructed from spaces of modular forms in Sect. 6) could be proved without relying on the main results of [33], and without using the potential diagonalizability of potentially Barsotti–Tate representations proved in [37] and [31], but rather just the explicit computations of potentially Barsotti–Tate deformation rings in Sect. 7. Of course, the results of Sect. 7 depend on [33] (via Theorem 7.1.1, which depends on [33] via [26]), so we will need to explain how to prove the results of both Sects. 6 and 7 (or at least enough of them to prove the main results of Sects. 8–10) without using these results. We will still make use of [34] in order to prove Theorem 7.1.1, but we remark that it should be possible to establish all of the results of Sect. 7 by directly extending the computations of Section 5 of [8] to the tame cuspidal case, so it should ultimately be possible to remove this dependence as well.

In fact, if we examine Sects. 6 and 7, we see that there are two things that need to be established: Theorem 7.2.1 (which only depends on the above items via Theorem 7.1.1), and the claim that if $\overline{\sigma }$ is a Serre weight and $M_\infty$ is constructed from the spaces of modular forms for a quaternion algebra using the Taylor–Wiles–Kisin method as in Sect. 6.2, then the action of the universal lifting ring $R$ on $M_\infty (\overline{\sigma })$ factors through $R^{\psi ,\overline{\sigma }}$ (or in fact through its reduction mod $\varpi _E$). (We recall that this is not immediately deducible from local-global compatibility because of parity issues: if $\sigma$ denotes the algebraic representation of $\mathrm{GL }_2(O_{F_v})$ lifting $\overline{\sigma }$, then the weights of $\sigma$ may not satisfy the parity condition necessary for constructing a local system on the congruence quotients associated to our given quaternion algebra.) Note that we only need to prove these claims under the assumption that $\overline{\rho }$ is generic (in the local case), or that $\overline{\rho }|_{G_{F_v}}$ is generic for all $v|p$ (in the global case), and we will make this assumption from now on.

The claim about $M_\infty (\overline{\sigma })$ will follow from the results of Sect. 7, in the following fashion. Suppose first that $\overline{\sigma }\notin {\mathcal {D}}(\overline{\rho })$; we will show that $M_\infty (\overline{\sigma })=0$ by an argument similar to those used in [33]. To do this, note that by Lemma 4.5.2 of [33] (and its proof), in the Grothendieck group of mod $p$ representations of $\mathrm{GL }_2(k_v)$ we can write $\overline{\sigma }$ as a linear combination $\sum _\tau n_\tau \overline{\sigma }(\tau )$ where $\tau$ runs over the tame inertial types (allowing for the moment $\tau$ to be the trivial or small Steinberg type). Then if $e$ denotes the Hilbert–Samuel multiplicity of a coherent sheaf on $X^\psi _\infty$, we have

$$\begin{aligned} e(M_\infty (\overline{\sigma }))=\sum _\tau n_\tau e(M_\infty (\overline{\sigma }(\tau ))). \end{aligned}$$

By the argument of Lemma 4.3.9 of [33], and the irreducibility of the generic fibres of the deformation spaces (see the discussion below), $\sum _\tau n_\tau e(M_\infty$ $(\overline{\sigma }(\tau )))$ is a multiple of $\sum _\tau n_\tau e(R^{\psi ,\tau })$ (and the constant of proportionality is independent of $\overline{\sigma }$), so it suffices to check that this last quantity is zero.

To see this, note that the $e(R^{\psi ,\tau })$ are completely determined by Theorem 7.2.1 and the observation that since $\overline{\rho }$ is assumed generic, $R^{\psi ,\tau }=0$ if $\tau$ is the trivial or Steinberg type (this reduces to checking that the reduction mod $p$ of a semistable (possibly crystalline) Galois representation of Hodge type $0$ is not generic, which is clear). In fact, we see that $e(R^{\psi ,\tau })=|\mathrm{JH }(\overline{\sigma }(\tau ))\cap {\mathcal {D}}(\overline{\rho })|$. Applying the argument of the previous paragraph to all $\overline{\sigma }$ simultaneously, and noting that again by Lemma 4.5.2 of [33] (and its proof) the $e(M_\infty (\overline{\sigma }(\tau )))$ determine the $e(M_\infty (\overline{\sigma }))$, we see that there must be some constant $c$ such that for any $\overline{\sigma }$, we have $e(M_\infty (\overline{\sigma }))=c$ if $\overline{\sigma }\in {\mathcal {D}}(\overline{\rho })$, and $0$ otherwise. In particular, if $\overline{\sigma }\notin {\mathcal {D}}(\overline{\rho })$, then $M_\infty (\overline{\sigma })=0$, as required.

Now suppose that $\overline{\sigma }\in {\mathcal {D}}(\overline{\rho })$. Then by Proposition 3.5.1 there is a tame inertial type $\tau$ such that $\mathrm{JH }(\overline{\sigma }(\tau ))\cap {\mathcal {D}}(\overline{\rho })=\{\overline{\sigma }\}$, so by the result of the previous paragraph we have $M_\infty (\overline{\sigma }^\circ (\tau ))=M_\infty (\overline{\sigma })$ for any lattice $\sigma ^\circ (\tau )$ in $\sigma (\tau )$. Also, by Theorem 7.1.1, the reduction mod $\varpi _E$ of $R^{\psi ,\overline{\sigma }}$ is equal to the reduction mod $\varpi _E$ of $R^{\psi ,\tau }$, so it suffices to note that the action of $R$ on $M_\infty (\overline{\sigma }^\circ (\tau ))$ factors through $R^{\psi ,\tau }$.

It remains to prove Theorem 7.1.1. Examining the proof of [26, Thm. 5.5.4], we see that we have to establish the existence of a suitable globalization in the sense of Section 5.1 of [26]), and we have to avoid the use of Lemma 4.4.1 of [33] (the potential diagonalizability of potentially Barsotti–Tate representations.). (Note that we still appeal to Lemma 4.4.2 of [33], and that this is where we make use of [34].) However, Lemma 4.4.1 of [33] is only used to establish that certain patched spaces of modular forms are supported on every component of the generic fibre of each local deformation ring which we consider, and this will be automatic provided we know that these generic fibres are domains. By a standard base change argument, it suffices to know this after a quadratic base change, so we can reduce to the case of tame principal series types, which follows from Lemma 7.4.1 (the proof of which makes no use of Theorem 7.1.1).

It remains to check the existence of a suitable globalization, which amounts to checking Conjecture A.3 of [26], the existence of a potentially diagonalizable lift of $\overline{\rho }$. In the proof of [26, Thm. 5.5.4], this is done by appealing to results of [33], which use the potential diagonalizability of potentially Barsotti–Tate representations. However, as we are assuming that $\overline{\rho }$ is generic, if we choose some weight $\overline{\sigma }\in {\mathcal {D}}(\overline{\rho })$ then a crystalline lift of $\overline{\rho }$ of Hodge type $\overline{\sigma }$ is Fontaine–Laffaille and thus potentially diagonalizable, as required.

Remark B.1.1

We suspect that the above analysis would make it possible to remove the assumption that when $p=5$ the projective image of $\overline{\rho }(G_{F(\zeta _5)})$ is not isomorphic to $A_5$ from our main global theorems, but we have not attempted to check the details.

Remark B.1.2

In particular, if we apply the above analysis to Theorem 9.1.1, we see that we can reprove the main result of [32] without making use of the results on the components of Barsotti–Tate deformation rings proved in [31, 37], and without needing to make an ordinarity assumption. (Of course, this ordinarity assumption was already removed by [5].)

1.2 Appendix B.2: Deformation rings and the structure of lattices

The main theme of this paper is the close connection between the structure of the lattices in tame types for $\mathrm{GL }_2(F_v)$, and the structures of the corresponding potentially Barsotti–Tate deformation rings for generic residual representations. We believe that this connection is even tighter than we have been able to show; in particular, we suspect that it should be possible to use explicit descriptions of the deformation rings to recover Theorem 5.1.1 without making use of the results of [9].

Unfortunately, we have not been able to prove this, and our partial results are fragmentary. One difficulty is that in general there will be irregular weights occurring as Jordan–Hölder factors of the reductions of the lattices, and it would be necessary to have an explicit description of the deformation rings for non-generic residual representations. However, even in cases where all the Jordan–Hölder factors are regular, we were still unable to prove complete results. As an illustration of what we were able to prove, we have the following modest result, whose proof will illustrate the kind of arguments that we have in mind.

Let $p>3$ be prime, let $F_v$ be an unramified extension of ${\mathbb {Q}_p}$, let $\tau$ be a tame type, and consider the lattice $\sigma ^\circ _J(\tau )$ for some $J\in \mathcal {P}_\tau$.

Proposition B.2.1

Suppose that $\overline{\sigma }_{J_1}(\tau ), \overline{\sigma }_{J_2}(\tau )$ are two weights in adjacent layers of the cosocle filtration of $\overline{\sigma }^\circ _J(\tau )$, and that there exists a nonsplit extension between $\overline{\sigma }_{J_1}(\tau ), \overline{\sigma }_{J_2}(\tau )$ as $\mathrm{GL }_2(F_v)$-representations. Then the extension induced by the cosocle filtration of $\overline{\sigma }^\circ _J(\tau )$ is also nonsplit.

Proof

By an explicit (but tedious) computation “dual” to that used in the proof of Proposition 10.1.11, the existence of a nonsplit extension between $\overline{\sigma }_{J_1}(\tau ), \overline{\sigma }_{J_2}(\tau )$ implies that there is a semisimple generic representation $\overline{\rho }$ such that $\mathrm{JH }(\overline{\sigma }(\tau )) \cap {\mathcal {D}}(\overline{\rho })=\{\overline{\sigma }_{J_1}(\tau ),\overline{\sigma }_{J_2}(\tau )\}$. Since $\overline{\rho }$ is semisimple, it is easy to globalise it as the local mod $p$ representation corresponding to some CM Hilbert modular form, and the constructions of Sect. 6.5 then give a minimal fixed determinant patching functor $M^{\min }_\infty$ with unramified coefficients $\mathcal {O}$, indexed by $(F_v,\overline{\rho })$.

As was already noted in Subsect. 5.2, it follows from Lemmas 4.1.1 and 3.1.1 that $\sigma ^\circ _J(\tau )$ is defined over the ring of integers of an unramified extension of ${\mathbb {Q}_p}$, so extending scalars if necessary, we can assume that it is defined over $\mathcal {O}$. By Theorem 7.2.1, and the assumptions that $|\mathrm{JH }(\overline{\sigma }(\tau )) \cap {\mathcal {D}}(\overline{\rho })|=2$ and $\mathcal {O}$ is unramified, we see that the deformation space $X^\psi \bigl (\tau \bigr )$ is regular, so by Lemma 6.1.4 we see that $M^{\min }_\infty (\sigma _J^\circ (\tau ))$ is free of rank one over $X^\psi _\infty \bigl (\tau \bigr )$, and thus in particular $M^{\min }_\infty (\overline{\sigma }_J^\circ (\tau ))$ is free of rank one over $\overline{X}^\psi _\infty \bigl (\tau \bigr )$.

Now, if the extension between $\overline{\sigma }_{J_1}(\tau ), \overline{\sigma }_{J_2}(\tau )$ induced by the cosocle filtration on $\overline{\sigma }^\circ _J(\tau )$ were split, we would have

$$\begin{aligned} M^{\min }_\infty (\overline{\sigma }_J^\circ (\tau ))=M^{\min }_\infty (\overline{\sigma }_{J_1}(\tau ))\oplus M^{\min }_\infty (\overline{\sigma }_{J_1}(\tau )) \end{aligned}$$

which would not be free of rank one, which is a contradiction; so the extension must be nonsplit, as claimed.