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On mod $p$ local-global compatibility for$\text{GL}_{3}$ in the ordinary case

Published online by Cambridge University Press:  25 August 2017

Florian Herzig
Affiliation:
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON M5S 2E4, Canada email herzig@math.toronto.edu
Daniel Le
Affiliation:
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON M5S 2E4, Canada email le@math.toronto.edu
Stefano Morra
Affiliation:
Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, Case courrier 051, Place Eugène Bataillon, 34095 Montpellier Cedex, France email stefano.morra@umontpellier.fr
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Abstract

Suppose that $F/F^{+}$ is a CM extension of number fields in which the prime $p$ splits completely and every other prime is unramified. Fix a place $w|p$ of $F$. Suppose that $\overline{r}:\operatorname{Gal}(\overline{F}/F)\rightarrow \text{GL}_{3}(\overline{\mathbb{F}}_{p})$ is a continuous irreducible Galois representation such that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ is upper-triangular, maximally non-split, and generic. If $\overline{r}$ is automorphic, and some suitable technical conditions hold, we show that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ can be recovered from the $\text{GL}_{3}(F_{w})$-action on a space of mod $p$ automorphic forms on a compact unitary group. On the way we prove results about weights in Serre’s conjecture for $\overline{r}$, show the existence of an ordinary lifting of $\overline{r}$, and prove the freeness of certain Taylor–Wiles patched modules in this context. We also show the existence of many Galois representations $\overline{r}$ to which our main theorem applies.

Type
Research Article
Copyright
© The Authors 2017 

1 Introduction

1.1 Motivation and statement of main results

Suppose that $p$ is a prime and that $\overline{\unicode[STIX]{x1D70C}}:\operatorname{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})\rightarrow \operatorname{GL}_{n}(\overline{\mathbb{F}}_{p})$ is a continuous Galois representation. One would hope that there is a mod $p$ Langlands correspondence that associates to $\overline{\unicode[STIX]{x1D70C}}$ in a natural way a smooth representation $\unicode[STIX]{x1D6F1}(\overline{\unicode[STIX]{x1D70C}})$ of $\operatorname{GL}_{n}(\mathbb{Q}_{p})$ over $\overline{\mathbb{F}}_{p}$ (or maybe a packet of such representations), and similarly for $p$ -adic representations. Unfortunately, at this point, this is only known for $n\leqslant 2$ [Reference BreuilBre03], [Reference ColmezCol10]. But suppose now that $F/F^{+}$ is a CM extension of number fields in which $p$ splits completely, and fix a place $w|p$ . Even in the absence of a local mod $p$ Langlands correspondence for $n>2$ , given a global automorphic Galois representation $\overline{r}:\operatorname{Gal}(\overline{F}/F)\rightarrow \operatorname{GL}_{n}(\overline{\mathbb{F}}_{p})$ we can define smooth representations $\unicode[STIX]{x1D6F1}_{\text{glob}}(\overline{r})$ of $\operatorname{GL}_{n}(\mathbb{Q}_{p})$ over $\overline{\mathbb{F}}_{p}$ on spaces of mod $p$ automorphic forms on a definite unitary group, that serve as candidates for $\unicode[STIX]{x1D6F1}(\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})})$ (in the spirit of Emerton’s local-global compatibility [Reference EmertonEme11]; see also [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEGGPS16] in the $p$ -adic setting). It is not known whether $\unicode[STIX]{x1D6F1}_{\text{glob}}(\overline{r})$ depends only on $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ . The motivating question of this paper is opposite to this: do the candidate representations $\unicode[STIX]{x1D6F1}_{\text{glob}}(\overline{r})$ contain at least as much information as $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ ? We answer this question in the affirmative in many cases when $n=3$ .

We fix a finite extension $E/\mathbb{Q}_{p}$ with residue field $\mathbb{F}$ , and consider absolutely irreducible Galois representations $\overline{r}:\operatorname{Gal}(\overline{F}/F)\rightarrow \operatorname{GL}_{3}(\mathbb{F})$ . We assume moreover that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ is upper-triangular, maximally non-split, and generic. This means that

(1.1.1) $$\begin{eqnarray}\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}\sim \left(\begin{array}{@{}ccc@{}}\unicode[STIX]{x1D714}^{a+1}\text{nr}_{\unicode[STIX]{x1D707}_{2}} & \ast _{1} & \ast \\ & \unicode[STIX]{x1D714}^{b+1}\text{nr}_{\unicode[STIX]{x1D707}_{1}} & \ast _{2}\\ & & \unicode[STIX]{x1D714}^{c+1}\text{nr}_{\unicode[STIX]{x1D707}_{0}}\end{array}\right),\end{eqnarray}$$

the extensions $\ast _{1}$ , $\ast _{2}$ are non-split, and $a-b>2$ , $b-c>2$ , $a-c<p-3$ . (Here, $\unicode[STIX]{x1D714}$ is the mod $p$ cyclotomic character and $\text{nr}_{\unicode[STIX]{x1D707}}$ denotes the unramified character taking value $\unicode[STIX]{x1D707}\in \mathbb{F}^{\times }$ on geometric Frobenius elements.) It is not hard to see that once the diagonal characters are fixed, the isomorphism class of $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ is determined by an invariant $\text{FL}(\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})})$ that can take any value in $\mathbb{P}^{1}(\mathbb{F})\setminus \{\unicode[STIX]{x1D707}_{1}\}$ . (We normalize this invariant using Fontaine–Laffaille theory, see Definition 2.1.10.)

To explain our results, we briefly describe our global setup, referring to § 4 for details. Fix a unitary group $G_{/F^{+}}$ such that $G\times F\cong \operatorname{GL}_{3}$ and $G(F_{v}^{+})\cong U_{3}(\mathbb{R})$ for all $v|\infty$ . Choose a model ${\mathcal{G}}_{/{\mathcal{O}}_{F^{+}}}$ of $G$ such that ${\mathcal{G}}\times {\mathcal{O}}_{F_{v^{\prime }}^{+}}$ is reductive for all places $v^{\prime }$ of $F^{+}$ that split in $F$ . Let $v=w|_{F^{+}}$ . Choose a compact open subgroup $U=U^{v}\times {\mathcal{G}}({\mathcal{O}}_{F_{v}^{+}})\leqslant G(\mathbb{A}_{F^{+}}^{\infty })$ that is sufficiently small, and unramified at all places dividing $p$ . Let $V^{\prime }$ denote an irreducible smooth representation of $\prod _{v^{\prime }|p,v^{\prime }\neq v}{\mathcal{G}}({\mathcal{O}}_{F_{v^{\prime }}^{+}})$ over $\mathbb{F}$ determined by a highest weight in the lowest alcove, and let $\widetilde{V}^{\prime }$ denote a Weyl module of $\prod _{v^{\prime }|p,v^{\prime }\neq v}{\mathcal{G}}({\mathcal{O}}_{F_{v^{\prime }}^{+}})$ over ${\mathcal{O}}_{E}$ of the same highest weight, so $\widetilde{V}^{\prime }\otimes _{{\mathcal{O}}_{E}}\mathbb{F}\cong V^{\prime }$ . (Except for Theorem C below, the reader may assume for simplicity that $V^{\prime }=\mathbb{F}$ and $\widetilde{V}^{\prime }={\mathcal{O}}_{E}$ .) We can then define in the usual way spaces of mod $p$ automorphic forms $S(U^{v},V^{\prime })=\mathop{\varinjlim }\nolimits_{U_{v}\leqslant G(F_{v}^{+})}S(U^{v}U_{v},V^{\prime })$ and similarly $S(U^{v},\widetilde{V}^{\prime })$ that are smooth representations of $G(F_{v}^{+})\cong \operatorname{GL}_{3}(\mathbb{Q}_{p})$ (where this isomorphism depends on our chosen place $w|v$ ).

We fix a cofinite subset ${\mathcal{P}}$ of places $w^{\prime }\nmid p$ of $F$ that split over $F^{+}$ , such that $U$ is unramified at $w^{\prime }|_{F^{+}}$ , and such that $\overline{r}$ is unramified at $w^{\prime }$ . Then the abstract Hecke algebra $\mathbb{T}^{{\mathcal{P}}}$ generated over ${\mathcal{O}}_{E}$ by Hecke operators at all places in ${\mathcal{P}}$ acts on $S(U^{v},V^{\prime })$ and $S(U^{v},\widetilde{V}^{\prime })$ , commuting with the $\operatorname{GL}_{3}(\mathbb{Q}_{p})$ -actions. Moreover, $\overline{r}$ determines a maximal ideal $\mathfrak{m}_{\overline{r}}$ of $\mathbb{T}^{{\mathcal{P}}}$ . We assume that $\overline{r}$ is automorphic in this setup, which means that $S(U^{v},V^{\prime })[\mathfrak{m}_{\overline{r}}]\neq 0$ (or equivalently, $S(U^{v},V^{\prime })_{\mathfrak{m}_{\overline{r}}}\neq 0$ ). These $\operatorname{GL}_{3}(\mathbb{Q}_{p})$ -representations, $S(U^{v},V^{\prime })[\mathfrak{m}_{\overline{r}}]$ , are the natural candidates $\unicode[STIX]{x1D6F1}_{\text{glob}}(\overline{r})$ that we mentioned above, at least if the level $U^{v}$ is chosen optimally.

It is a consequence of earlier work of the first-named author and Breuil [Reference Breuil and HerzigBH15] that the $\operatorname{GL}_{3}(\mathbb{Q}_{p})$ -representation $S(U^{v},V^{\prime })[\mathfrak{m}_{\overline{r}}]$ determines the ordered triple of diagonal characters $(\unicode[STIX]{x1D714}^{a+1}\text{nr}_{\unicode[STIX]{x1D707}_{2}},\unicode[STIX]{x1D714}^{b+1}\text{nr}_{\unicode[STIX]{x1D707}_{1}},\unicode[STIX]{x1D714}^{c+1}\text{nr}_{\unicode[STIX]{x1D707}_{0}})$ of $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ . In fact, the triple $(a,b,c)$ can be recovered from the (ordinary part of the) $\operatorname{GL}_{3}(\mathbb{Z}_{p})$ -socle, i.e., the Serre weights of $\overline{r}$ , by [Reference Gee and GeraghtyGG12] and the $\unicode[STIX]{x1D707}_{i}\in \mathbb{F}^{\times }$ are determined by the Hecke action at $p$ on the $\operatorname{GL}_{3}(\mathbb{Z}_{p})$ -socle. It therefore remains for us to show that $S(U^{v},V^{\prime })[\mathfrak{m}_{\overline{r}}]$ determines the invariant $\text{FL}(\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})})$ . (We note that the representation $\unicode[STIX]{x1D6F1}(\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})})^{\text{ord}}$ of [Reference Breuil and HerzigBH15] does not contain this information.)

Let $I$ denote the Iwahori subgroup of $\operatorname{GL}_{3}(\mathbb{Z}_{p})$ , which is the preimage of the upper-triangular matrices $B(\mathbb{F}_{p})$ in $\operatorname{GL}_{3}(\mathbb{F}_{p})$ . If $V$ is a representation of $\operatorname{GL}_{3}(\mathbb{Z}_{p})$ over ${\mathcal{O}}_{E}$ and $a_{i}\in \mathbb{Z}$ we write

$$\begin{eqnarray}V^{I,(a_{2},a_{1},a_{0})}\stackrel{\text{def}}{=}\operatorname{Hom}_{I}({\mathcal{O}}_{E}(\widetilde{\unicode[STIX]{x1D714}}^{a_{2}}\otimes \widetilde{\unicode[STIX]{x1D714}}^{a_{1}}\otimes \widetilde{\unicode[STIX]{x1D714}}^{a_{0}}),V),\end{eqnarray}$$

where the character in the domain denotes the inflation to $I$ of the homomorphism $B(\mathbb{F}_{p})\rightarrow {\mathcal{O}}_{E}^{\times }$ , $\Bigl(\!\begin{smallmatrix}x & y & z\\ & u & v\\ & & w\end{smallmatrix}\!\Bigr)\mapsto \widetilde{x}^{a_{2}}\widetilde{u}^{a_{1}}\widetilde{w}^{a_{0}}$ . If $V$ is even a representation of $\operatorname{GL}_{3}(\mathbb{Q}_{p})$ , then $V^{I,(a_{2},a_{1},a_{0})}$ affords an action of $U_{p}$ -operators $U_{1}$ , $U_{2}$ (see (3.1.10)). Define also $\unicode[STIX]{x1D6F1}\stackrel{\text{def}}{=}\Bigl(\!\begin{smallmatrix} & 1 & \\ & & 1\\ p & & \end{smallmatrix}\!\Bigr)$ , which sends $V^{I,(a_{2},a_{1},a_{0})}$ to $V^{I,(a_{1},a_{0},a_{2})}$ .

Finally, and crucially, we define explicit group algebra operators $S$ , $S^{\prime }\in \mathbb{F}[\operatorname{GL}_{3}(\mathbb{F}_{p})]$ , see (4.5.1). We can now state our first main theorem.

Theorem A (Theorem 4.5.2).

We make the following additional assumptions:

  1. (i) $\text{FL}(\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})})\not \in \{0,\infty \}$ ;

  2. (ii) the ${\mathcal{O}}_{E}$ -dual of $S(U^{v},\widetilde{V}^{\prime })_{\mathfrak{m}_{\overline{r}}}^{I,(-b,-c,-a)}$ is free over $\mathbb{T}$ , where $\mathbb{T}$ denotes the ${\mathcal{O}}_{E}$ -subalgebra of $\operatorname{End}\big(S(U^{v},\widetilde{V}^{\prime })_{\mathfrak{m}_{\overline{r}}}^{I,(-b,-c,-a)}\big)$ generated by $\mathbb{T}^{{\mathcal{P}}}$ , $U_{1}$ , and $U_{2}$ .

Then we have the equality

$$\begin{eqnarray}S^{\prime }\circ \unicode[STIX]{x1D6F1}=(-1)^{a-b}\cdot \frac{b-c}{a-b}\cdot \text{FL}(\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})})\cdot S\end{eqnarray}$$

of maps

$$\begin{eqnarray}S(U^{v},V^{\prime })[\mathfrak{m}_{\overline{r}}]^{I,(-b,-c,-a)}[U_{1},U_{2}]\rightarrow S(U^{v},V^{\prime })[\mathfrak{m}_{\overline{r}}]^{I,(-c-1,-b,-a+1)}.\end{eqnarray}$$

Moreover, these maps are injective with non-zero domain. In particular, $\text{FL}(\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})})$ is determined by the smooth $\operatorname{GL}_{3}(\mathbb{Q}_{p})$ -representation $S(U^{v},V^{\prime })[\mathfrak{m}_{\overline{r}}]$ .

The first assumption is related to the surprising fact that the $\operatorname{GL}_{3}(\mathbb{Z}_{p})$ -socle of $S(U^{v},V^{\prime })[\mathfrak{m}_{\overline{r}}]$ changes for the exceptional two invariants, see Theorem D below. (Incidentally, this means that even in the exceptional cases $\text{FL}(\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})})$ is determined by $S(U^{v},V^{\prime })[\mathfrak{m}_{\overline{r}}]$ .)

We show that the second assumption is often a consequence of the first assumption. The second assumption is an analogue of Mazur’s ‘mod $p$ multiplicity one’ result, and thus our result may be of independent interest. We have not tried to optimize our hypotheses, the most stringent of which is that $U$ may be taken to be unramified at all finite places.

Theorem B (Theorem 5.1.1).

Assume hypotheses (i)–(ix) in § 5.1. If $\text{FL}(\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})})\neq \infty$ , then assumption (ii) in Theorem A holds.

In fact we show that for any value of $\text{FL}(\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})})$ either the second assumption or its dual holds (see Remark 5.3.6).

We also show, using results of [Reference Emerton and GeeEG14], that for any given local Galois representation as in (1.1.1) we can construct a globalization to which Theorem B applies.

Theorem C (Theorem 5.3.7).

Suppose that $\overline{\unicode[STIX]{x1D70C}}:\operatorname{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})\rightarrow \operatorname{GL}_{3}(\mathbb{F})$ is upper-triangular, maximally non-split, and generic. Then, after possibly replacing $\mathbb{F}$ by a finite extension, there exist a CM field $F$ , a Galois representation $\overline{r}:\operatorname{Gal}(\overline{F}/F)\rightarrow \operatorname{GL}_{3}(\mathbb{F})$ , a place $w|p$ of $F$ , groups $G_{/F^{+}}$ and ${\mathcal{G}}_{/{\mathcal{O}}_{F^{+}}}$ , and a compact open subgroup $U^{v}$ (where $v=w|_{F^{+}}$ ) satisfying all hypotheses of the setup in § 5.1 such that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}\cong \overline{\unicode[STIX]{x1D70C}}$ . In particular, if $\text{FL}(\overline{\unicode[STIX]{x1D70C}})\not \in \{0,\infty \}$ , Theorem A applies to $\overline{r}$ .

As a by-product of our methods we almost completely determine the set of Serre weights of $\overline{r}$ . Here, the set $W_{w}(\overline{r})$ is defined to be the set of irreducible $\operatorname{GL}_{3}(\mathbb{Z}_{p})$ -representations whose duals occur in the $\operatorname{GL}_{3}(\mathbb{Z}_{p})$ -socle of $S(U^{v},V^{\prime })_{\mathfrak{m}_{\overline{r}}}$ (for some $U^{v}$ and ${\mathcal{P}}$ as above). See § 4.2 for our notation for Serre weights.

Theorem D (Theorem 4.4.1).

Keep the assumptions on $\overline{r}$ that precede Theorem A above.

  1. (i) If $\text{FL}(\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})})\notin \{0,\infty \}$ we have

    $$\begin{eqnarray}\displaystyle & \{F(a-1,b,c+1)\}\subseteq W_{w}(\overline{r})\subseteq \{F(a-1,b,c+1),F(c+p-1,b,a-p+1)\}. & \displaystyle \nonumber\end{eqnarray}$$
  2. (ii) If $\text{FL}(\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})})=\infty$ we have

    $$\begin{eqnarray}\displaystyle & & \displaystyle \{F(a-1,b,c+1),F(a,c,b-p+1)\}\nonumber\\ \displaystyle & & \displaystyle \qquad \subseteq W_{w}(\overline{r})\subseteq \{F(a-1,b,c+1),F(c+p-1,b,a-p+1),F(a,c,b-p+1)\}.\nonumber\end{eqnarray}$$
  3. (iii) If $\text{FL}(\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})})=0$ we have

    $$\begin{eqnarray}\displaystyle & & \displaystyle \{F(a-1,b,c+1),F(b+p-1,a,c)\}\nonumber\\ \displaystyle & & \displaystyle \qquad \subseteq W_{w}(\overline{r})\subseteq \{F(a-1,b,c+1),F(c+p-1,b,a-p+1),F(b+p-1,a,c)\}.\nonumber\end{eqnarray}$$

This is one of the first Serre weight results in dimension 3. It was completed in early 2014, before the recent progress of [Reference Le, Hung, Levin and MorraLHLM15] (on Serre weights in dimension 3 in the generic semisimple case, using different methods). As we said above, the dependence on $\text{FL}(\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})})$ in this theorem was unexpected, as there were no explicit Serre weight conjectures in the literature that apply to non-semisimple $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ .

As a consequence of this theorem we also show the existence of an automorphic lift $r$ of $\overline{r}$ such that $r|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ is upper-triangular. It is in the same spirit as the main results of [Reference Barnet-Lamb, Gee and GeraghtyBLGG12] (which concerned two-dimensional representations).

Theorem E (Corollary 4.4.4).

In the setting of Theorem D, $\overline{r}$ has an automorphic lift $r:\operatorname{Gal}(\overline{F}/F)\rightarrow \operatorname{GL}_{3}({\mathcal{O}}_{E})$ (after possibly enlarging $E$ ) such that $r|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ is crystalline and ordinary of Hodge–Tate weights $\{-a-1,-b-1,-c-1\}$ .

1.2 Methods used

Theorems A and B generalize earlier work of Breuil and Diamond [Reference Breuil and DiamondBD14] which treated the case of two-dimensional Galois representations of $\operatorname{Gal}(\overline{F}/F)$ , where $F$ is totally real and $p$ is unramified in $F$ . We follow the same general strategy: we lift the Hecke eigenvalues of $\overline{r}$ to a well-chosen type in characteristic zero, use classical local-global compatibility at $p$ , and then study carefully how both the Galois-side and the $\operatorname{GL}_{3}$ -side reduce modulo $p$ . However, it is significantly more difficult to carry out this strategy in the $\operatorname{GL}_{3}$ -setting.

We first prove the upper bound in Theorem D by lifting to various types in characteristic zero and using integral $p$ -adic Hodge theory to reduce modulo $p$ . This is more involved in dimension 3, since we are no longer in the potentially Barsotti–Tate setting. We crucially use results of Caruso to filter our Breuil module (corresponding to $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ ) according to the socle filtration on $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ , see Proposition 2.3.5.

The following theorem is our key local result on the Galois-side. Our chosen type is a tame principal series that contains in its reduction modulo $p$ all elements of $W_{w}(\overline{r})$ (unlike in [Reference Breuil and DiamondBD14], where the intersection always consisted of one element). In contrast to [Reference Breuil and DiamondBD14] we get away with a rough classification of the strongly divisible module corresponding to $\unicode[STIX]{x1D70C}$ . (We do not need to determine Frobenius and monodromy operators.) We also note that the relevant information on the Galois-side is independent of the Hodge filtration, so that we can transfer this information to the $\operatorname{GL}_{3}$ -side using classical local-global compatibility.

Theorem F (Theorem 2.5.1).

Let $\unicode[STIX]{x1D70C}:\operatorname{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})\rightarrow \text{GL}_{3}({\mathcal{O}}_{E})$ be a potentially semistable $p$ -adic Galois representation of Hodge–Tate weights $\{-2,-1,0\}$ and inertial type $\widetilde{\unicode[STIX]{x1D714}}^{a}\oplus \widetilde{\unicode[STIX]{x1D714}}^{b}\oplus \widetilde{\unicode[STIX]{x1D714}}^{c}$ . Assume that the residual representation $\overline{\unicode[STIX]{x1D70C}}:\operatorname{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})\rightarrow \text{GL}_{3}(\mathbb{F})$ is upper-triangular, maximally non-split, and generic as in (1.1.1). Let $\unicode[STIX]{x1D706}\in {\mathcal{O}}_{E}$ be the Frobenius eigenvalue on $\text{D}_{\text{st}}^{\mathbb{Q}_{p},2}(\unicode[STIX]{x1D70C})^{I_{\mathbb{Q}_{p}}=\widetilde{\unicode[STIX]{x1D714}}^{b}}$ . Then the Fontaine–Laffaille invariant of $\overline{\unicode[STIX]{x1D70C}}$ is given by

$$\begin{eqnarray}\text{FL}(\overline{\unicode[STIX]{x1D70C}})=\operatorname{red}(p\unicode[STIX]{x1D706}^{-1}),\end{eqnarray}$$

where $\operatorname{red}$ denotes the specialization map $\mathbb{P}^{1}({\mathcal{O}}_{E})\rightarrow \mathbb{P}^{1}(\mathbb{F})$ .

On the $\operatorname{GL}_{3}$ -side our main innovation consists of the explicit group algebra operators  $S$ $S^{\prime }$ . The analogues of these operators for $\operatorname{GL}_{2}$ show up in various contexts (see, for example, [Reference PaskunasPas07, Lemma 4.1], [Reference Breuil and PaškūnasBP12, Lemma 2.7], [Reference BreuilBre11, § 4], and [Reference Breuil and DiamondBD14, Proposition 2.6.1]). Our proof is specific to $\operatorname{GL}_{3}$ . It would be interesting to find a more conceptual explanation for them. See also Question 3.1.3 for a further discussion of such operators.

Proposition G (Proposition 3.1.2).

  1. (i) There is a unique non-split extension of irreducible $\operatorname{GL}_{3}(\mathbb{F}_{p})$ -representations

    $$\begin{eqnarray}0\rightarrow F(-c-1,-b,-a+1)\rightarrow V\rightarrow F(-b+p-1,-c,-a)\rightarrow 0\end{eqnarray}$$
    and $S$ induces an isomorphism $S:\,V^{I,(-b,-c,-a)}\stackrel{{\sim}}{\longrightarrow }V^{I,(-c-1,-b,-a+1)}$ of one-dimensional vector spaces.
  2. (ii) There is a unique non-split extension of irreducible $\operatorname{GL}_{3}(\mathbb{F}_{p})$ -representations

    $$\begin{eqnarray}0\rightarrow F(-c-1,-b,-a+1)\rightarrow V\rightarrow F(-c,-a,-b-p+1)\rightarrow 0\end{eqnarray}$$
    and $S^{\prime }$ induces an isomorphism $S^{\prime }:\,V^{I,(-c,-a,-b)}\stackrel{{\sim}}{\longrightarrow }V^{I,(-c-1,-b,-a+1)}$ of one-dimensional vector spaces.

The result concerning reduction modulo $p$ on the $\operatorname{GL}_{3}$ -side is comparatively easier, see Proposition 3.2.2.

By combining the above results we deduce Theorem A. We note that assumption (ii) is needed for lifting elements of $S(U^{v},V^{\prime })[\mathfrak{m}_{\overline{r}}]^{I,(-b,-c,-a)}[U_{1},U_{2}]$ to suitable Iwahori eigenvectors in characteristic zero. The $U_{i}$ -operators allow us to deal with the possible presence of the shadow weight $F(c+p-1,b,a-p+1)$ in Theorem D. (The term ‘shadow weight’ is defined in [Reference Emerton, Gee and HerzigEGH13, § 6], and more generally in [Reference Gee, Herzig and SavittGHS16, §§ 1.5 and 7.2].) Namely, if $v\in S(U^{v},V^{\prime })$ $[\mathfrak{m}_{\overline{r}}]^{I,(-b,-c,-a)}[U_{1},U_{2}]$ is non-zero, we show that it generates the representation of Proposition G(i) under the $\operatorname{GL}_{3}(\mathbb{Z}_{p})$ -action. Similar comments apply to $\unicode[STIX]{x1D6F1}v$ . Proposition G then allows us to deduce that the maps in Theorem A are well defined and injective. (We refer to Remark 4.5.8 for variations on assumption (ii). The stronger assumptions appearing in Remark 4.5.9 are analogous to the multiplicity one conditions appearing in [Reference Breuil and DiamondBD14].)

Interestingly, the argument proving Theorem A also lets us deduce the hardest part of Theorem D, namely the existence of the shadow weights $F(a,c,b-p+1)$ , $F(b+p-1,a,c)$ in the two exceptional cases. After [Reference Emerton, Gee and HerzigEGH13], this is only the second result in the literature proving the existence of shadow weights. (Again this precedes [Reference Le, Hung, Levin and MorraLHLM15].)

Finally, we establish Theorem B. As in [Reference Breuil and DiamondBD14] our method relies on the Taylor–Wiles method. However, as we do not know whether our local deformation ring at $p$ is formally smooth (which in any case should be false if our chosen type intersects $W_{w}(\overline{r})$ in more than one element) we cannot directly apply Diamond’s method [Reference DiamondDia97]. Instead we use the patched modules of [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEGGPS16] that live over the universal deformation space at $p$ and use ideas of [Reference Emerton, Gee and SavittEGS15] and [Reference LeLe15]. See Theorem 5.2.3 for our freeness result at infinite level from which we deduce Theorem B. Similarly to above, we add $U_{p}$ -operators in order to deal with the possible presence of the shadow weight $F(c+p-1,b,a-p+1)$ .

1.3 Notation

Let $\overline{\mathbb{Q}}$ be an algebraic closure of $\mathbb{Q}$ . All number fields $F/\mathbb{Q}$ will be considered as subfields of $\overline{\mathbb{Q}}$ and we write $G_{F}\stackrel{\text{def}}{=}\operatorname{Gal}(\overline{\mathbb{Q}}/F)$ to denote the absolute Galois group of $F$ . For any rational prime $\ell \in \mathbb{Q}$ , we fix algebraic closures $\overline{\mathbb{Q}}_{\ell }$ of $\mathbb{Q}_{\ell }$ and embeddings $\overline{\mathbb{Q}}{\hookrightarrow}\overline{\mathbb{Q}}_{\ell }$ (hence inclusions $G_{\mathbb{Q}_{\ell }}{\hookrightarrow}G_{\mathbb{Q}}$ ). The residue field of $\overline{\mathbb{Q}}_{\ell }$ , which is an algebraic closure of $\mathbb{F}_{\ell }$ , is denoted by $\overline{\mathbb{F}}_{\ell }$ . As above, all algebraic extensions of $\mathbb{Q}_{\ell }$ , $\mathbb{F}_{\ell }$ will be considered as subfields of the fixed algebraic closures $\overline{\mathbb{Q}}_{\ell }$ , $\overline{\mathbb{F}}_{\ell }$ .

Let $k/\mathbb{F}_{p}$ be a finite extension of degree $f\geqslant 1$ , and let $K_{0}\stackrel{\text{def}}{=}W(k)[1/p]$ be the unramified extension of $\mathbb{Q}_{p}$ of degree $f$ . Suppose that $e\geqslant 1$ is any divisor of $p^{f}-1$ . (Starting in § 2.4 we will assume $e=p^{f}-1$ . However, in the appendix it will be convenient to allow more general $e$ , in particular $e=1$ .) We consider the Eisenstein polynomial $E(u)\stackrel{\text{def}}{=}u^{e}+p\in \mathbb{Q}_{p}[u]$ and fix a root $\unicode[STIX]{x1D71B}=\sqrt[e]{-p}\in \overline{\mathbb{Q}}_{p}$ . Let $K\stackrel{\text{def}}{=}K_{0}(\unicode[STIX]{x1D71B})$ , a tamely totally ramified cyclic extension of $K_{0}$ of degree $e$ with uniformizer $\unicode[STIX]{x1D71B}$ .

Let $E$ be a finite extension of $\mathbb{Q}_{p}$ . We write ${\mathcal{O}}_{E}$ for its ring of integers, $\mathbb{F}$ for its residue field and $\unicode[STIX]{x1D71B}_{E}\in {\mathcal{O}}_{E}$ to denote a uniformizer. We always assume that $K\subseteq E$ .

The choice of $\unicode[STIX]{x1D71B}\in K$ provides us with a homomorphism

$$\begin{eqnarray}\displaystyle \begin{array}{@{}rcl@{}}\displaystyle \widetilde{\unicode[STIX]{x1D714}}_{\unicode[STIX]{x1D71B}}:\operatorname{Gal}(K/\mathbb{Q}_{p})\ & \longrightarrow \ & W(k)^{\times }\\ \displaystyle g\ & \longmapsto \ & \displaystyle \frac{g(\unicode[STIX]{x1D71B})}{\unicode[STIX]{x1D71B}}\end{array} & & \displaystyle \nonumber\end{eqnarray}$$

whose reduction modulo $p$ will be denoted by $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}$ . Note that the inclusion $k\subseteq \mathbb{F}$ induced by $K\subseteq E$ provides us with a niveau $f$ fundamental character $\unicode[STIX]{x1D714}_{f}:\operatorname{Gal}(K/K_{0})\rightarrow \mathbb{F}^{\times }$ , namely $\unicode[STIX]{x1D714}_{f}=\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}|_{\operatorname{Gal}(K/K_{0})}$ .

We denote by $\unicode[STIX]{x1D714}:G_{\mathbb{Q}_{p}}\rightarrow \mathbb{F}_{p}^{\times }$ the mod $p$ cyclotomic character, so $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{1}$ .

Write $\unicode[STIX]{x1D711}$ for the $p$ -power Frobenius on $k$ . We recall the standard idempotent elements $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D70E}}\in k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F}$ defined for $\unicode[STIX]{x1D70E}\in \operatorname{Hom}(k,\mathbb{F})$ , which verify $(\unicode[STIX]{x1D711}\,\otimes \,1)(\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D70E}})=\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D70E}\circ \unicode[STIX]{x1D711}^{-1}}$ and $(\unicode[STIX]{x1D706}\,\otimes \,1)\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D70E}}=(1\,\otimes \,\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D706}))\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D70E}}$ . We write $\widehat{\unicode[STIX]{x1D716}}_{\unicode[STIX]{x1D70E}}\in W(k)\otimes _{\mathbb{Z}_{p}}{\mathcal{O}}_{E}$ for the standard idempotent elements; they reduce to $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D70E}}$ modulo  $\unicode[STIX]{x1D71B}_{E}$ .

Our convention on Hodge–Tate weights is that the cyclotomic character $\unicode[STIX]{x1D700}:G_{\mathbb{Q}_{p}}\rightarrow \mathbb{Q}_{p}^{\times }$ has Hodge–Tate weight $-1$ .

Given a potentially semistable $p$ -adic representation $\unicode[STIX]{x1D70C}:G_{\mathbb{Q}_{p}}\rightarrow \operatorname{GL}_{n}(E)$ , we write $\operatorname{WD}(\unicode[STIX]{x1D70C})$ to denote the associated Weil–Deligne representation as defined in [Reference Conrad, Diamond and TaylorCDT99, Appendix B.1]. Therefore, $\unicode[STIX]{x1D70C}\mapsto \operatorname{WD}(\unicode[STIX]{x1D70C})$ is a covariant functor. We refer to $\operatorname{WD}(\unicode[STIX]{x1D70C})|_{I_{\mathbb{Q}_{p}}}$ as the inertial type associated to $\unicode[STIX]{x1D70C}$ .

2 The local Galois side

In this section we analyze the local Galois side. In particular, we establish Theorem F of the introduction.

2.1 Fontaine–Laffaille invariant

Let $\overline{\unicode[STIX]{x1D70C}}:G_{\mathbb{Q}_{p}}\rightarrow \operatorname{GL}_{3}(\mathbb{F})$ be a continuous Galois representation. We assume that $\overline{\unicode[STIX]{x1D70C}}$ is maximally non-split meaning that $\overline{\unicode[STIX]{x1D70C}}$ is uniserial and the graded pieces in its socle filtration are at most one dimensional over $\mathbb{F}$ . (Recall that a finite length module is uniserial if it has a unique composition series.) In other words,

(2.1.1) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D70C}}\sim \left(\begin{array}{@{}ccc@{}}\unicode[STIX]{x1D714}^{a_{2}+1}\text{nr}_{\unicode[STIX]{x1D707}_{2}} & \ast _{1} & \ast \\ & \unicode[STIX]{x1D714}^{a_{1}+1}\text{nr}_{\unicode[STIX]{x1D707}_{1}} & \ast _{2}\\ & & \unicode[STIX]{x1D714}^{a_{0}+1}\text{nr}_{\unicode[STIX]{x1D707}_{0}}\end{array}\right)\end{eqnarray}$$

for some $a_{i}\in \mathbb{Z}$ , $\unicode[STIX]{x1D707}_{i}\in \mathbb{F}^{\times }$ and where $\ast _{1},\ast _{2}$ are non-split. Here, $\text{nr}_{\unicode[STIX]{x1D707}}:G_{\mathbb{Q}_{p}}\rightarrow \mathbb{F}^{\times }$ denotes the unramified character taking the value $\unicode[STIX]{x1D707}$ on a geometric Frobenius element of $G_{\mathbb{Q}_{p}}$ .

2.1.1 Preliminaries on Fontaine–Laffaille theory

We begin by briefly recalling the theory of Fontaine–Laffaille modules with $\mathbb{F}$ -coefficients and its relation with mod  $p$ Galois representations.

A Fontaine–Laffaille module $(M,\operatorname{Fil}^{\bullet }M,\unicode[STIX]{x1D719}_{\bullet })$ over $k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F}$ is the datum of

  1. (i) a finite free $k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F}$ -module $M$ ;

  2. (ii) a decreasing filtration $\{\operatorname{Fil}^{j}M\}_{j\in \mathbb{Z}}$ on $M$ by $k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F}$ -submodules such that $\operatorname{Fil}^{0}M=M$ and $\operatorname{Fil}^{p-1}M=0$ ;

  3. (iii) a $\unicode[STIX]{x1D711}$ -semilinear isomorphism $\unicode[STIX]{x1D719}_{\bullet }:\text{gr}^{\bullet }M\rightarrow M$ .

Defining the morphisms in the obvious way, we obtain the abelian category $\mathbb{F}\text{-}{\mathcal{F}}{\mathcal{L}}^{[0,p-2]}$ of Fontaine–Laffaille modules over $k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F}$ . Given a Fontaine–Laffaille module $M$ and $\unicode[STIX]{x1D70E}\in \operatorname{Hom}(k,\mathbb{F})$ , we define the Hodge–Tate weights of $M$ with respect to $\unicode[STIX]{x1D70E}$ :

$$\begin{eqnarray}\text{HT}_{\unicode[STIX]{x1D70E}}(M)\stackrel{\text{def}}{=}\biggl\{i\in \mathbb{N}:\dim _{\mathbb{F}}\biggl(\frac{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D70E}}\operatorname{Fil}^{i}M}{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D70E}}\operatorname{Fil}^{i+1}M}\biggr)\neq 0\biggr\}.\end{eqnarray}$$

In the remainder of this paper we focus on Fontaine–Laffaille modules with parallel Hodge–Tate weights, i.e. we assume that for all $i\in \mathbb{N}$ , the submodules $\operatorname{Fil}^{i}M$ are free over $k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F}$ .

Definition 2.1.2. Let $M$ be a Fontaine–Laffaille module with parallel Hodge–Tate weights. A $k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F}$ -basis $\text{}\underline{e}=(e_{0},\ldots ,e_{n-1})$ on $M$ is compatible with the Hodge filtration if for all $i\in \mathbb{N}$ there exists $j_{i}\in \mathbb{N}$ such that $\operatorname{Fil}^{i}M=\sum _{j=j_{i}}^{n}(k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F})\cdot e_{j}$ .

Note that if the graded pieces of the Hodge filtration have rank at most one, then any two compatible bases on $M$ are related by a lower triangular matrix in $\operatorname{GL}_{n}(k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F})$ .

Given a Fontaine–Laffaille module and a compatible basis $\text{}\underline{e}$ , it is convenient to describe the Frobenius action via a matrix $\text{Mat}_{\text{}\underline{e}}(\unicode[STIX]{x1D719}_{\bullet })\in \operatorname{GL}_{n}(k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F})$ , defined in the obvious way using the principal symbols $(\text{gr}(e_{0}),\ldots ,\text{gr}(e_{n-1}))$ as a basis of $\text{gr}^{\bullet }M$ .

Theorem 2.1.3. There is an exact, fully faithful contravariant functor

$$\begin{eqnarray}\text{T}_{\text{cris}}^{\ast }:\,\mathbb{F}\text{-}{\mathcal{F}}{\mathcal{L}}^{[0,p-2]}\rightarrow \text{Rep}_{\mathbb{ F}}(G_{K_{0}})\end{eqnarray}$$

which is moreover compatible with base change: if $K_{0}^{\prime }/K_{0}$ is finite unramified, with residue field $k^{\prime }/k$ , then

$$\begin{eqnarray}\text{T}_{\text{cris}}^{\ast }(M\otimes _{k}k^{\prime })\cong \text{T}_{\text{cris}}^{\ast }(M)|_{G_{K_{0}^{\prime }}}.\end{eqnarray}$$

Also, the essential image of $\text{T}_{\text{cris}}^{\ast }$ is closed under subquotients.

Proof. The statement with $\mathbb{F}_{p}$ -coefficients is in [Reference Fontaine and LaffailleFL82, Théorème 6.1]; its analogue with $\mathbb{F}$ -coefficient is a formal argument which is left to the reader (cf. also [Reference Gao and LiuGL14, Theorem 2.2.1]).◻

Lemma 2.1.4. Let $\overline{\unicode[STIX]{x1D70C}}:G_{\mathbb{Q}_{p}}\rightarrow \operatorname{GL}_{3}(\mathbb{F})$ be as in (2.1.1). If the integers $a_{i}$ verify $a_{1}-a_{0}>1,a_{2}-a_{1}>1$ and $p-2>a_{2}-a_{0}$ then $\overline{\unicode[STIX]{x1D70C}}\,\otimes \,\unicode[STIX]{x1D714}^{-a_{0}}$ is Fontaine–Laffaille, i.e. it is in the essential image of $\text{T}_{\text{cris}}^{\ast }$ .

Proof. This follows, for example, from [Reference Gee and GeraghtyGG12, Lemma 3.1.5]. ◻

In order to obtain the main results on Serre weights (§ 4.4) and local-global compatibility (§ 4.5), we must assume a stronger genericity condition on the integers $a_{i}$ .

Definition 2.1.5. We say that a maximally non-split Galois representation $\overline{\unicode[STIX]{x1D70C}}:G_{\mathbb{Q}_{p}}\rightarrow \operatorname{GL}_{3}(\mathbb{F})$ as in (2.1.1) is generic if the triple $(a_{2},a_{1},a_{0})$ satisfies the condition

(2.1.6) $$\begin{eqnarray}a_{1}-a_{0}>2,\quad a_{2}-a_{1}>2,\quad p-3>a_{2}-a_{0}.\end{eqnarray}$$

2.1.2 The Fontaine–Laffaille invariant

Let $\overline{\unicode[STIX]{x1D70C}}:G_{\mathbb{Q}_{p}}\rightarrow \operatorname{GL}_{3}(\mathbb{F})$ be as in Definition 2.1.5. By Lemma 2.1.4 there is a Fontaine–Laffaille module $M$ with Hodge–Tate weights $\{1,a_{1}-a_{0}+1,a_{2}-a_{0}+1\}$ such that $\text{T}_{\text{cris}}^{\ast }(M)\cong \overline{\unicode[STIX]{x1D70C}}\,\otimes \,\unicode[STIX]{x1D714}^{-a_{0}}$ and which is moreover endowed with a filtration by Fontaine–Laffaille submodules $0\subsetneq M_{0}\subsetneq M_{1}\subsetneq M_{2}=M$ induced via $\text{T}_{\text{cris}}^{\ast }$ from the cosocle filtration on $\overline{\unicode[STIX]{x1D70C}}$ (cf. Theorem 2.1.3).

Lemma 2.1.7. Let $M\in \mathbb{F}\text{-}{\mathcal{F}}{\mathcal{L}}^{[0,p-2]}$ be such that $\text{T}_{\text{cris}}^{\ast }(M)\cong \overline{\unicode[STIX]{x1D70C}}\,\otimes \,\unicode[STIX]{x1D714}^{-a_{0}}$ . There exists a basis $\text{}\underline{e}=(e_{0},e_{1},e_{2})$ of $M$ such that

$$\begin{eqnarray}M_{i}\cap \operatorname{Fil}^{a_{i}-a_{0}+1}M=\mathbb{F}\cdot e_{i}\end{eqnarray}$$

for all $i\in \{0,1,2\}$ .

Proof. This follows by noting that $M_{i}\,\cap \,\operatorname{Fil}^{a_{i}-a_{0}+1}M=\operatorname{Fil}^{a_{i}-a_{0}+1}M_{i}$ and that $M_{i}$ has Hodge–Tate weights $a_{j}-a_{0}+1$ with $0\leqslant j\leqslant i$ .◻

Note that a basis $\text{}\underline{e}$ as in Lemma 2.1.7 is compatible with the Hodge filtration. From Lemma 2.1.7 we deduce a useful observation on the Frobenius action on $M$ .

Corollary 2.1.8. Let $\overline{\unicode[STIX]{x1D70C}}$ , $M$ , $\text{}\underline{e}$ be as in Lemma 2.1.7. Then

$$\begin{eqnarray}\text{Mat}_{\text{}\underline{e}}(\unicode[STIX]{x1D719}_{\bullet })=\left(\begin{array}{@{}ccc@{}}\unicode[STIX]{x1D707}_{0} & \unicode[STIX]{x1D6FC}_{01} & \unicode[STIX]{x1D6FC}_{02}\\ & \unicode[STIX]{x1D707}_{1} & \unicode[STIX]{x1D6FC}_{12}\\ & & \unicode[STIX]{x1D707}_{2}\end{array}\right)\in \operatorname{GL}_{3}(\mathbb{F}),\end{eqnarray}$$

where, moreover, $\unicode[STIX]{x1D6FC}_{01},\unicode[STIX]{x1D6FC}_{12}\in \mathbb{F}^{\times }$ .

Proof. This follows from the lemma, by recalling that the Fontaine–Laffaille module associated to a $G_{\mathbb{Q}_{p}}$ -character $\unicode[STIX]{x1D714}^{r}\text{nr}_{\unicode[STIX]{x1D707}}$ has Hodge–Tate weight $r$ and $\unicode[STIX]{x1D719}=\unicode[STIX]{x1D707}$ . Note that $\unicode[STIX]{x1D6FC}_{01},\unicode[STIX]{x1D6FC}_{12}\neq 0$ as $\overline{\unicode[STIX]{x1D70C}}$ is maximally non-split.◻

Conversely, we also note that any such matrix defines a Fontaine–Laffaille module whose associated Galois representation is maximally non-split as in (2.1.1).

The Fontaine–Laffaille invariant $\text{FL}(\overline{\unicode[STIX]{x1D70C}})$ associated to $\overline{\unicode[STIX]{x1D70C}}$ is defined in terms of $\text{Mat}_{\text{}\underline{e}}(\unicode[STIX]{x1D719}_{\bullet })$ .

Lemma 2.1.9. Keep the hypotheses of Corollary 2.1.8. The element $\unicode[STIX]{x1D6FC}_{02}/\unicode[STIX]{x1D6FC}_{01}\unicode[STIX]{x1D6FC}_{12}$ deduced from $\text{Mat}_{\text{}\underline{e}}(\unicode[STIX]{x1D719}_{\bullet })$ does not depend on the choice of the basis $\text{}\underline{e}$ .

Proof. By Lemma 2.1.7 any other such basis is of the form $\unicode[STIX]{x1D6FD}_{i}e_{i}$ for $\unicode[STIX]{x1D6FD}_{i}\in \mathbb{F}^{\times }$ . The lemma follows.◻

Definition 2.1.10. Let $\overline{\unicode[STIX]{x1D70C}}:G_{\mathbb{Q}_{p}}\rightarrow \operatorname{GL}_{3}(\mathbb{F})$ be maximally non-split and generic as in Definition 2.1.5. Let $M$ be the Fontaine–Laffaille module associated to $\overline{\unicode[STIX]{x1D70C}}\,\otimes \,\unicode[STIX]{x1D714}^{-a_{0}}$ , $\text{}\underline{e}=(e_{0},e_{1},e_{2})$ a basis of $M$ as in Lemma 2.1.7 and let

$$\begin{eqnarray}\text{Mat}_{\text{}\underline{e}}(\unicode[STIX]{x1D719}_{\bullet })=\left(\begin{array}{@{}ccc@{}}\unicode[STIX]{x1D707}_{0} & \unicode[STIX]{x1D6FC}_{01} & \unicode[STIX]{x1D6FC}_{02}\\ & \unicode[STIX]{x1D707}_{1} & \unicode[STIX]{x1D6FC}_{12}\\ & & \unicode[STIX]{x1D707}_{2}\end{array}\right)\end{eqnarray}$$

be the matrix of the Frobenius action on $M$ .

The Fontaine–Laffaille invariant associated to $\overline{\unicode[STIX]{x1D70C}}$ is defined as

$$\begin{eqnarray}\text{FL}(\overline{\unicode[STIX]{x1D70C}})\stackrel{\text{def}}{=}\frac{\det \left(\begin{array}{@{}cc@{}}\unicode[STIX]{x1D6FC}_{01} & \unicode[STIX]{x1D6FC}_{02}\\ \unicode[STIX]{x1D707}_{1} & \unicode[STIX]{x1D6FC}_{12}\end{array}\right)}{-\unicode[STIX]{x1D6FC}_{02}}\in \mathbb{P}^{1}(\mathbb{F})=\mathbb{F}\cup \{\infty \}.\end{eqnarray}$$

By Lemma 2.1.9 it is well defined.

Remark 2.1.11. Let $\overline{\unicode[STIX]{x1D70C}}$ be maximally non-split as in (2.1.1). The isomorphism class of $\overline{\unicode[STIX]{x1D70C}}$ is determined by the diagonal characters $\unicode[STIX]{x1D714}^{a_{i}+1}\text{nr}_{\unicode[STIX]{x1D707}_{i}}$ and the Fontaine–Laffaille invariant $\text{FL}(\overline{\unicode[STIX]{x1D70C}})$ . Note that $\text{FL}(\overline{\unicode[STIX]{x1D70C}})$ can take any value in $\mathbb{P}^{1}(\mathbb{F})$ except for $\unicode[STIX]{x1D707}_{1}$ . (Similarly, a maximally non-split Galois representation $\overline{\unicode[STIX]{x1D70C}}:G_{\mathbb{Q}_{p}}\rightarrow \operatorname{GL}_{n}(\mathbb{F})$ is determined by the diagonal characters and $\binom{n-1}{2}$ invariants.)

Remark 2.1.12. Note that in the situation of Definition 2.1.10, if $\mathbb{F}^{\prime }/\mathbb{F}$ is a finite field extension, then $\text{FL}(\overline{\unicode[STIX]{x1D70C}}\otimes _{\mathbb{F}}\mathbb{F}^{\prime })=\text{FL}(\overline{\unicode[STIX]{x1D70C}})$ .

Remark 2.1.13. We leave it to the reader to show that the Fontaine–Laffaille module associated to $\overline{\unicode[STIX]{x1D70C}}^{\vee }\,\otimes \,\unicode[STIX]{x1D714}^{a_{2}+2}$ is described by

$$\begin{eqnarray}\left(\begin{array}{@{}ccc@{}} & & 1\\ & 1 & \\ 1 & & \end{array}\right)\cdot ^{t}\text{Mat}_{\text{}\underline{e}}(\unicode[STIX]{x1D719}_{\bullet })^{-1}\cdot \left(\begin{array}{@{}ccc@{}} & & 1\\ & 1 & \\ 1 & & \end{array}\right).\end{eqnarray}$$

As a consequence, $\text{FL}(\overline{\unicode[STIX]{x1D70C}}^{\vee })=\text{FL}(\overline{\unicode[STIX]{x1D70C}})^{-1}$ .

2.2 $p$ -adic Hodge theory

This section mainly consists of a review of some integral $p$ -adic Hodge theory, although many of the results are not available in the literature in the form or generality that we need.

In the first subsection (§ 2.2.1) we define the categories of mod $p$ objects we are going to work with (Breuil modules with descent data, étale $\unicode[STIX]{x1D711}$ -modules, etc.). Moreover, we obtain a key result, Corollary 2.2.2, which provides a criterion for deciding when a given Breuil module with descent data and a Fontaine–Laffaille module have isomorphic Galois representations.

The second subsection (§ 2.2.2) is of a more technical nature. On the one hand we make two of the functors from § 2.2.1 (relating Breuil, Fontaine–Laffaille and étale $\unicode[STIX]{x1D711}$ -modules) more explicit. We also provide a useful change-of-basis result for a Breuil module with descent data.

All missing proofs of this section are contained in § A.5. Our rationale is to state in this section all the results we need to prove our main results on the Galois side, and to relegate technical details to the appendix.

2.2.1 Breuil modules with tame descent data

Let $K^{\prime }\subseteq K_{0}$ be a subfield containing $\mathbb{Q}_{p}$ . The residual Breuil ring $\overline{S}\stackrel{\text{def}}{=}(k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F})[u]/(u^{ep})$ is equipped with an action of $\text{Gal}(K/K^{\prime })$ by $k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F}$ -semilinear automorphisms. Explicitly if $g\in \text{Gal}(K/K^{\prime })$ and $a\in k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F}$ , we have

$$\begin{eqnarray}\widehat{g}(au^{i})\stackrel{\text{def}}{=}(g\otimes 1)(a)\cdot (\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}(g)\otimes 1)^{i}u^{i}.\end{eqnarray}$$

If $\overline{\unicode[STIX]{x1D712}}:\text{Gal}(K/K^{\prime })\rightarrow \mathbb{F}^{\times }$ is a character, we write $\overline{S}_{\overline{\unicode[STIX]{x1D712}}}$ to denote the $\overline{\unicode[STIX]{x1D712}}$ -isotypical component of $\overline{S}$ for the action of $\operatorname{Gal}(K/K^{\prime })$ .

We recall that $\overline{S}$ is equipped with an $k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F}$ -linear derivation $N\stackrel{\text{def}}{=}-u(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}u)$ and with a semilinear Frobenius $\unicode[STIX]{x1D711}$ defined by $u\mapsto u^{p}$ (semilinear with respect to the arithmetic Frobenius $\unicode[STIX]{x1D711}\,\otimes \,1$ on $k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F}$ ), which moreover commute with the action of $\operatorname{Gal}(K/K^{\prime })$ on $\overline{S}$ .

Fix $r\in \{0,\ldots ,p-2\}$ . A Breuil module of weight  $r$  with descent data from  $K$  to  $K^{\prime }$ is a quintuple $({\mathcal{M}},\operatorname{Fil}^{r}{\mathcal{M}},\unicode[STIX]{x1D711}_{r},N,\{\widehat{g}\})$ , consisting of the following data.

  1. (i) A $\bar{S}$ -module ${\mathcal{M}}$ that is finite free.

  2. (ii) A $\bar{S}$ -submodule $\operatorname{Fil}^{r}{\mathcal{M}}$ of ${\mathcal{M}}$ , verifying $u^{er}{\mathcal{M}}\subseteq \operatorname{Fil}^{r}{\mathcal{M}}$ .

  3. (iii) A morphism $\unicode[STIX]{x1D711}_{r}:\operatorname{Fil}^{r}{\mathcal{M}}\rightarrow {\mathcal{M}}$ which is $\unicode[STIX]{x1D711}$ -semilinear and whose image generates ${\mathcal{M}}$ .

  4. (iv) An operator $N:{\mathcal{M}}\rightarrow {\mathcal{M}}$ that is $k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F}$ -linear and satisfies certain axioms (see the beginning of [Reference Emerton, Gee and HerzigEGH13, § 3.2]).

  5. (v) An action of $\text{Gal}(K/K^{\prime })$ on ${\mathcal{M}}$ by automorphisms $\widehat{g}$ which are semilinear with respect to the Galois action on $\overline{S}$ and which preserve $\operatorname{Fil}^{r}{\mathcal{M}}$ and commute with $\unicode[STIX]{x1D711}_{r}$ and $N$ .

A morphism of Breuil modules is an $\overline{S}$ -linear morphism which is compatible, in the evident sense, with the additional structures.

We write $\mathbb{F}\text{-}\operatorname{BrMod}_{\text{dd}}^{r}$ to denote the category of Breuil modules of weight $r$ with descent data from $K$ to $K^{\prime }$ ; the field $K^{\prime }$ will always be clear from the context (and will be specified in case of ambiguities). As we did for the coefficient ring $\overline{S}$ , given a character $\overline{\unicode[STIX]{x1D712}}:\text{Gal}(K/K^{\prime })\rightarrow \mathbb{F}^{\times }$ we write ${\mathcal{M}}_{\overline{\unicode[STIX]{x1D712}}}$ , $(\operatorname{Fil}^{r}{\mathcal{M}})_{\overline{\unicode[STIX]{x1D712}}}$ to denote the $\overline{\unicode[STIX]{x1D712}}$ -isotypical component of ${\mathcal{M}}$ , $\operatorname{Fil}^{r}{\mathcal{M}}$ respectively.

We remark that the category $\mathbb{F}\text{-}\operatorname{BrMod}_{\text{dd}}^{r}$ is additive and admits kernels and cokernels (cf. [Reference CarusoCar11, Théorème 4.2.4] and the Remarque following it). In particular a complex

$$\begin{eqnarray}0\rightarrow {\mathcal{M}}_{1}\stackrel{f_{1}}{\rightarrow }{\mathcal{M}}_{2}\stackrel{f_{2}}{\rightarrow }{\mathcal{M}}_{3}\rightarrow 0\end{eqnarray}$$

in $\mathbb{F}\text{-}\operatorname{BrMod}_{\text{dd}}^{r}$ is exact if the morphisms $f_{i}$ induce exact sequences on the underlying $\overline{S}$ -modules ${\mathcal{M}}_{j}$ and $\operatorname{Fil}^{r}{\mathcal{M}}_{j}$ ( $j\in \{0,1,2\}$ ). This endows $\mathbb{F}\text{-}\operatorname{BrMod}_{\text{dd}}^{r}$ with the structure of an exact category (see Proposition 2.3.4 below).

We recall that we have an exact, faithful, contravariant functor

$$\begin{eqnarray}\displaystyle \text{T}_{\text{st}}^{\ast }:\mathbb{F}\text{-}\operatorname{BrMod}_{\text{dd}}^{r} & \rightarrow & \displaystyle \text{Rep}_{\mathbb{F}}(G_{K^{\prime }})\nonumber\\ \displaystyle {\mathcal{M}} & \mapsto & \displaystyle \text{T}_{\text{st}}^{\ast }({\mathcal{M}})\stackrel{\text{def}}{=}\text{Hom}({\mathcal{M}},\widehat{A}),\nonumber\end{eqnarray}$$

where $\widehat{A}$ is a certain period ring and homomorphisms respect all structures (cf. § A.3). We have $\dim _{\mathbb{F}}\,\text{T}_{\text{st}}^{\ast }({\mathcal{M}})=\operatorname{rank}_{\overline{S}}{\mathcal{M}}$ (cf. [Reference CarusoCar11, Théorème 4.2.4], and the Remarque following it; see also [Reference Emerton, Gee and HerzigEGH13, Lemma 3.2.2]).

We will be mainly concerned with the covariant version $\text{T}_{\text{st}}^{r}({\mathcal{M}})\stackrel{\text{def}}{=}(\text{T}_{\text{st}}^{\ast }({\mathcal{M}}))^{\vee }\,\otimes \,\unicode[STIX]{x1D714}^{r}$ of the functor $\text{T}_{\text{st}}^{\ast }$ above. We remark that this is compatible with the notion of duality ${\mathcal{M}}\mapsto {\mathcal{M}}^{\ast }$ on $\mathbb{F}\text{-}\operatorname{BrMod}_{\text{dd}}^{r}$ recalled in § A.4, namely $\text{T}_{\text{st}}^{r}({\mathcal{M}})\cong \text{T}_{\text{st}}^{\ast }({\mathcal{M}}^{\ast })$ , cf. the discussion before [Reference Emerton, Gee and HerzigEGH13, Corollary 3.2.9].

We now move to the categories of étale $\unicode[STIX]{x1D711}$ -modules. In the remainder of this subsection we will assume $K^{\prime }=K_{0}$ , i.e. we will only consider descent data from $K$ to $K_{0}$ .

We fix the field of norms $k((\text{}\underline{\unicode[STIX]{x1D71B}}))$ associated to a suitable Kummer extension $K_{\infty }$ of $K$ (see § A.1 for its precise definition). It is endowed with a $p$ -power Frobenius endomorphism and with an action of $\operatorname{Gal}(K/K_{0})$ .

An étale  $(\unicode[STIX]{x1D719},k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F}((\text{}\underline{\unicode[STIX]{x1D71B}})))$ -module with descent data is the datum of a finite free $k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F}((\text{}\underline{\unicode[STIX]{x1D71B}}))$ -module $\mathfrak{M}$ endowed with a semilinear injective Frobenius endomorphism $\unicode[STIX]{x1D719}:\mathfrak{M}\rightarrow \mathfrak{M}$ and a semilinear action of $\operatorname{Gal}(K/K_{0})$ , commuting with $\unicode[STIX]{x1D719}$ . We write $\mathbb{F}\text{-}\mathfrak{Mod}_{k((\text{}\underline{\unicode[STIX]{x1D71B}})),\text{dd}}$ to denote the category of étale $(\unicode[STIX]{x1D719},k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F}((\text{}\underline{\unicode[STIX]{x1D71B}})))$ -modules with descent data. In the special case when $e=1$ this category is denoted by $\mathbb{F}\text{-}\mathfrak{Mod}_{k((\text{}\underline{p}))}$ . (In this case $k((\text{}\underline{p}))$ is also the field of norms associated to a suitable Kummer extension $(K_{0})_{\infty }/K_{0}$ .) We remark that $k((\text{}\underline{\unicode[STIX]{x1D71B}}))/k((\text{}\underline{p}))$ is a cyclic extension of degree $e$ with Galois group $\operatorname{Gal}(K/K_{0})$ , and we write $k((\text{}\underline{p}))^{s}$ to denote a choice of separable closure of $k((\text{}\underline{p}))$ containing $k((\text{}\underline{\unicode[STIX]{x1D71B}}))$ . For details, see § A.1.

Finally, recall the category of Fontaine–Laffaille modules $\mathbb{F}\text{-}{\mathcal{F}}{\mathcal{L}}^{[0,p-2]}$ over $k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F}$ (defined in § 2.1.1) and the category $\mathbb{F}\text{-}\operatorname{BrMod}_{\text{dd}}^{r}$ of Breuil modules of weight $r$ with descent data from $K$ to $K_{0}$ (defined in § 2.2.1).

The relations between the categories introduced so far are summarized in the following proposition. Its proof, as well as the definition of the functors $M_{k((\text{}\underline{\unicode[STIX]{x1D71B}}))}$ , ${\mathcal{F}}$ , … can be found in Appendix A.

Proposition 2.2.1. Let $0\leqslant r\leqslant p-2$ . We have the following commutative diagram:

where the descent data is from $K$ to $K_{0}$ . Moreover, the functor $\text{res}\circ \text{T}_{\text{cris}}^{\ast }$ is fully faithful.

We record the following immediate, yet crucial, corollary.

Corollary 2.2.2. Let $0\leqslant r\leqslant p-2$ and let ${\mathcal{M}}$ , $M$ be objects in $\mathbb{F}\text{-}\operatorname{BrMod}_{\text{dd}}^{r}$ and $\mathbb{F}\text{-}{\mathcal{F}}{\mathcal{L}}^{[0,p-2]}$ respectively. Assume that $\text{T}_{\text{st}}^{\ast }({\mathcal{M}})$ is Fontaine–Laffaille. If

$$\begin{eqnarray}M_{k((\text{}\underline{\unicode[STIX]{x1D71B}}))}({\mathcal{M}})\cong {\mathcal{F}}(M)\otimes _{k((\text{}\underline{p}))}k((\text{}\underline{\unicode[STIX]{x1D71B}}))\end{eqnarray}$$

then one has an isomorphism of $G_{K_{0}}$ -representations

$$\begin{eqnarray}\text{T}_{\text{st}}^{\ast }({\mathcal{M}})\cong \text{T}_{\text{cris}}^{\ast }(M).\end{eqnarray}$$

Let us explain the role that Corollary 2.2.2 plays in the proof of our main theorem on the Galois side (Theorem 2.5.1). Thus suppose that $\unicode[STIX]{x1D70C}$ is potentially semistable of Hodge–Tate weights $\{-2,-1,0\}$ with reduction $\overline{\unicode[STIX]{x1D70C}}$ that is maximally non-split and generic as in Definition 2.1.5. Associated to $\unicode[STIX]{x1D70C}$ is a strongly divisible module $\widehat{{\mathcal{M}}}$ , whose reduction ${\mathcal{M}}\in \mathbb{F}\text{-}\operatorname{BrMod}_{\text{dd}}^{r}$ has Galois representation $\overline{\unicode[STIX]{x1D70C}}$ . Corollary 2.2.2 will allow us to compute the Fontaine–Laffaille module $M$ associated to $\overline{\unicode[STIX]{x1D70C}}$ – and hence the invariant $\text{FL}(\overline{\unicode[STIX]{x1D70C}})$ – in terms of $\widehat{{\mathcal{M}}}$ , so in terms of $\unicode[STIX]{x1D70C}$ . In fact, we will start with ${\mathcal{M}}$ in the top left corner of the diagram and then go around it in a clockwise sense: first computing $M_{k((\text{}\underline{\unicode[STIX]{x1D71B}}))}({\mathcal{M}})$ , then descending it to $\mathbb{F}\text{-}\mathfrak{Mod}_{k((\text{}\underline{p}))}$ and finally to $\mathbb{F}\text{-}{\mathcal{F}}{\mathcal{L}}^{[0,p-2]}$ . To obtain the precise result, Theorem 2.5.1, we will moreover need more information about $\widehat{{\mathcal{M}}}$ , and this will be obtained in § 2.4.

2.2.2 Linear algebra with descent data

We continue to assume that $K^{\prime }=K_{0}$ . It will be convenient to introduce bases $\text{}\underline{e}$ (respectively generating sets $\text{}\underline{f}$ ) of a Breuil module ${\mathcal{M}}$ with descent data (respectively of $\operatorname{Fil}^{r}{\mathcal{M}}$ ) that are compatible with the action of $\operatorname{Gal}(K/K_{0})$ , and to describe $\operatorname{Fil}^{r}{\mathcal{M}}$ (respectively $\unicode[STIX]{x1D711}_{r}$ ) by matrices with respect to $\text{}\underline{e}$ , $\text{}\underline{f}$ . We then use this formalism to make the functors $M_{k((\text{}\underline{\unicode[STIX]{x1D71B}}))}$ , ${\mathcal{F}}$ appearing in the diagram of Proposition 2.2.1 more explicit, as well as we obtain a change-of-basis result for a Breuil module with descent data. Proofs can be found in § A.5.

Definition 2.2.3. We say that a Breuil module ${\mathcal{M}}\in \mathbb{F}\text{-}\operatorname{BrMod}_{\text{dd}}^{r}$ is of type $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{a_{0}}\oplus \ldots \oplus \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{a_{n-1}}$ (where $a_{i}\in \mathbb{Z}$ ) if ${\mathcal{M}}/u{\mathcal{M}}\cong \bigoplus _{i=0}^{n-1}(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{a_{i}}\,\otimes \,1)$ as $(k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F})[\operatorname{Gal}(K/K_{0})]$ -modules. Equivalently, ${\mathcal{M}}$ has an $\overline{S}$ -basis $(e_{0},\ldots ,e_{n-1})$ such that $\widehat{g}e_{i}=(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{a_{i}}(g)\,\otimes \,1)e_{i}$ for all $i$ and all $g\in \operatorname{Gal}(K/K_{0})$ . We call such a basis a framed basis of  ${\mathcal{M}}$ .

If ${\mathcal{M}}$ is of type $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{a_{0}}\oplus \ldots \oplus \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{a_{n-1}}$ we say that $(f_{0},\ldots ,f_{n-1})$ is a framed system of generators of  $\operatorname{Fil}^{r}{\mathcal{M}}$ if $\operatorname{Fil}^{r}{\mathcal{M}}=\sum _{i=0}^{n-1}\overline{S}\cdot f_{i}$ and $\widehat{g}f_{i}=(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{p^{-1}a_{i}}(g)\,\otimes \,1)f_{i}$ for all $i$ and all $g\in \operatorname{Gal}(K/K_{0})$ .

To justify the claim implicit in this definition, choose an $\overline{S}/u$ -basis $(\overline{e}_{1},\ldots ,\overline{e}_{n-1})$ of ${\mathcal{M}}/u{\mathcal{M}}$ such that $\widehat{g}\cdot \overline{e}_{i}=(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{a_{i}}(g)\,\otimes \,1)\overline{e}_{i}$ for all $i$ and all $g\in \operatorname{Gal}(K/K_{0})$ . Since $(k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F})[\operatorname{Gal}(K/K_{0})]$ is a semisimple (commutative) ring we can pick a $(k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F})[\operatorname{Gal}(K/K_{0})]$ -linear splitting of ${\mathcal{M}}{\twoheadrightarrow}{\mathcal{M}}/u{\mathcal{M}}$ and hence find $e_{i}\in {\mathcal{M}}$ lifting $\overline{e}_{i}$ such that $\widehat{g}e_{i}=(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{a_{i}}(g)\,\otimes \,1)e_{i}$ . By Nakayama’s lemma, the $e_{i}$ form an $\overline{S}$ -basis of ${\mathcal{M}}$ .

The notion of a framed basis (respectively a framed system of generators) depends on an ordering of the integers $a_{i}$ . It will always be clear from the context which ordering we use.

Lemma 2.2.4. If ${\mathcal{M}}\in \mathbb{F}\text{-}\operatorname{BrMod}_{\text{dd}}^{r}$ is of type $\bigoplus _{i=0}^{n-1}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{a_{i}}$ , then $\operatorname{Fil}^{r}{\mathcal{M}}$ admits a framed system of generators.

Definition 2.2.5. Let ${\mathcal{M}}\in \mathbb{F}\text{-}\operatorname{BrMod}_{\text{dd}}^{r}$ be of type $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{a_{0}}\oplus \ldots \oplus \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{a_{n-1}}$ . Let $\text{}\underline{e}$ , $\text{}\underline{f}$ be a framed basis and a framed system of generators of ${\mathcal{M}}$ , $\operatorname{Fil}^{r}{\mathcal{M}}$ respectively. The matrix of the filtration, with respect to $\text{}\underline{e},\text{}\underline{f}$ , is the element $\text{Mat}_{\text{}\underline{e},\text{}\underline{f}}(\operatorname{Fil}^{r}{\mathcal{M}})\in \text{M}_{n}(\overline{S})$ verifying

$$\begin{eqnarray}\text{}\underline{f}=\text{}\underline{e}\cdot \text{Mat}_{\text{}\underline{e},\text{}\underline{f}}(\operatorname{Fil}^{r}{\mathcal{M}}).\end{eqnarray}$$

We define the matrix of Frobenius with respect to $\text{}\underline{e}$ , $\text{}\underline{f}$ as the element $\text{Mat}_{\text{}\underline{e},\text{}\underline{f}}(\unicode[STIX]{x1D711}_{r})\in \text{GL}_{n}(\overline{S})$ characterized by

$$\begin{eqnarray}\unicode[STIX]{x1D711}_{r}(\text{}\underline{f})=\text{}\underline{e}\cdot \text{Mat}_{\text{}\underline{e},\text{}\underline{f}}(\unicode[STIX]{x1D711}_{r}).\end{eqnarray}$$

As we require $\text{}\underline{e},\text{}\underline{f}$ to be framed, the coefficients $\text{Mat}_{\text{}\underline{e},\text{}\underline{f}}(\operatorname{Fil}^{r}{\mathcal{M}})_{ij}$ verify the following conditions:

$$\begin{eqnarray}\text{Mat}_{\text{}\underline{e},\text{}\underline{f}}(\operatorname{Fil}^{r}{\mathcal{M}})_{ij}\in \overline{S}_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{p^{-1}a_{j}-a_{i}}}.\end{eqnarray}$$

Concretely $\text{Mat}_{\text{}\underline{e},\text{}\underline{f}}(\operatorname{Fil}^{r}{\mathcal{M}})_{ij}=u^{[p^{-1}a_{j}-a_{i}]}s_{ij}$ , where $[x]\in \{0,\ldots ,e-1\}$ is defined by $[x]\equiv x\,\text{mod}\,e$ for $x\in \mathbb{Z}/e\mathbb{Z}$ and $s_{ij}\in \overline{S}_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{0}}=k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F}[u^{e}]/(u^{ep})$ .

On the other hand, $\text{Mat}_{\text{}\underline{e},\text{}\underline{f}}(\unicode[STIX]{x1D711}_{r})\in \text{GL}_{n}^{\Box }(\overline{S})$ , where

$$\begin{eqnarray}\operatorname{GL}_{n}^{\Box }(\overline{S})\stackrel{\text{def}}{=}\{A\in \operatorname{GL}_{n}(\overline{S}):A_{ij}\in \overline{S}_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{a_{j}-a_{i}}}\,\,\text{for all}\,\,0\leqslant i,j\leqslant n-1\}.\end{eqnarray}$$

Lemma 2.2.6. Let ${\mathcal{M}}$ be a Breuil module of type $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{a_{0}}\oplus \ldots \oplus \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{a_{n-1}}$ , and let $\text{}\underline{e}$ be a framed basis of ${\mathcal{M}}$ and $\text{}\underline{f}$ a framed system of generators of $\operatorname{Fil}^{r}{\mathcal{M}}$ , respectively. Let $V\stackrel{\text{def}}{=}\text{Mat}_{\text{}\underline{e},\text{}\underline{f}}(\operatorname{Fil}^{r}{\mathcal{M}})\in \text{M}_{n}(\overline{S})$ and $A\stackrel{\text{def}}{=}\text{Mat}_{\text{}\underline{e},\text{}\underline{f}}(\unicode[STIX]{x1D711}_{r})\in \text{GL}_{n}^{\Box }(\overline{S})$ . Then there exists a basis $\text{}\underline{\mathfrak{e}}$ of $M_{k((\text{}\underline{\unicode[STIX]{x1D71B}}))}({\mathcal{M}}^{\ast })$ with $\widehat{g}\cdot \mathfrak{e}_{i}=(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{-p^{-1}a_{i}}(g)\,\otimes \,1)\mathfrak{e}_{i}$ for all $i$ and $g\in \operatorname{Gal}(K/K_{0})$ , such that $\text{Mat}_{\text{}\underline{\mathfrak{e}}}(\unicode[STIX]{x1D719})\in \text{M}_{n}(k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F}[[\text{}\underline{\unicode[STIX]{x1D71B}}]])$ is given by any chosen lift of $^{t}V\cdot ^{t}A^{-1}\in \text{M}_{n}(\overline{S})$ via the morphism $k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F}[[\text{}\underline{\unicode[STIX]{x1D71B}}]]{\twoheadrightarrow}\overline{S}$ sending $\sum \unicode[STIX]{x1D706}_{i}\text{}\underline{\unicode[STIX]{x1D71B}}^{i}$ to $\sum \unicode[STIX]{x1D706}_{i}u^{i}$ and such that $(\text{Mat}_{\text{}\underline{\mathfrak{e}}}(\unicode[STIX]{x1D719}))_{ij}\in (k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F}[[\text{}\underline{\unicode[STIX]{x1D71B}}]])_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{p^{-1}a_{i}-a_{j}}}$ for all $0\leqslant i,j\leqslant n-1$ .

Lemma 2.2.7. Let $M\in \mathbb{F}\text{-}{\mathcal{F}}{\mathcal{L}}^{[0,p-2]}$ be a rank $n$ Fontaine–Laffaille module with parallel Hodge–Tate weights $0\leqslant m_{0}\leqslant \ldots \leqslant m_{n-1}\leqslant p-2$ (counted with multiplicities).

Let $\text{}\underline{e}=(e_{0},\ldots ,e_{n-1})$ be a $k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F}$ -basis of $M_{i}$ that is compatible with the Hodge filtration $\text{Fil}^{\bullet }M$ , and let $F\in \operatorname{GL}_{n}(k\,\otimes _{\mathbb{F}_{p}}\,\mathbb{F})$ be the associated matrix of the Frobenius $\unicode[STIX]{x1D719}_{\bullet }:\operatorname{gr}^{\bullet }M\xrightarrow[{}]{\,\sim \,}M$ .

Then there exists a basis $\text{}\underline{\mathfrak{e}}$ of $\mathfrak{M}\stackrel{\text{def}}{=}{\mathcal{F}}(M)$ in $\mathbb{F}\text{-}\mathfrak{Mod}_{k((\text{}\underline{p}))}$ such that the Frobenius $\unicode[STIX]{x1D719}$ on $\mathfrak{M}$ is described by

$$\begin{eqnarray}\text{Mat}_{\text{}\underline{\mathfrak{e}}}(\unicode[STIX]{x1D719})=\text{Diag}(\text{}\underline{p}^{m_{0}}\ldots \text{}\underline{p}^{m_{n-1}})F.\end{eqnarray}$$

Lemma 2.2.8. Let ${\mathcal{M}}\in \mathbb{F}\text{-}\operatorname{BrMod}_{\text{dd}}^{r}$ be a Breuil module of type $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{a_{0}}\oplus \ldots \oplus \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{a_{n-1}}$ and let $\text{}\underline{e}$ , $\text{}\underline{f}$ be a framed basis of ${\mathcal{M}}$ and a framed system of generators of $\operatorname{Fil}^{r}{\mathcal{M}}$ respectively.

Write $V\stackrel{\text{def}}{=}\text{Mat}_{\text{}\underline{e},\text{}\underline{f}}(\operatorname{Fil}^{r}{\mathcal{M}})$ , $A\stackrel{\text{def}}{=}\text{Mat}_{\text{}\underline{e},\text{}\underline{f}}(\unicode[STIX]{x1D711}_{r})$ . Assume that there exist elements $V^{\prime }\in \text{M}_{n}(\overline{S})$ , $B\in \operatorname{GL}_{n}(\overline{S})$ such that $V_{ij}^{\prime }\in \overline{S}_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{p^{-1}a_{j}-a_{i}}}$ , $B_{ij}\in \overline{S}_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D71B}}^{p^{-1}(a_{j}-a_{i})}}$ and

(2.2.9) $$\begin{eqnarray}AV^{\prime }\equiv VB\hspace{0.2em}{\rm mod}\hspace{0.2em}u^{e(r+1)}.\end{eqnarray}$$

Then $\text{}\underline{e}^{\prime }\stackrel{\text{def}}{=}\text{}\underline{e}\cdot A$ is a framed basis of ${\mathcal{M}}$ , $\text{}\underline{f}^{\prime }\stackrel{\text{def}}{=}\text{}\underline{e}^{\prime }\cdot V^{\prime }$ is a framed system of generators for $\operatorname{Fil}^{r}{\mathcal{M}}$ , and $\text{Mat}_{\text{}\underline{e}^{\prime },\text{}\underline{f}^{\prime }}(\unicode[STIX]{x1D711}_{r})=\unicode[STIX]{x1D711}(B)$ .

2.3 Breuil submodules and Galois representations

In this subsection we discuss some preliminaries on subobjects and quotients in the category $\mathbb{F}\text{-}\operatorname{BrMod}_{\text{dd}}^{r}$ . Even though these notions are presumably well known to the experts, we did not find a suitable reference in the literature. The main result, Proposition 2.3.5, is a slight improvement of a result of Caruso [Reference CarusoCar11]. All missing proofs of this subsection are contained in § A.6.

In what follows, we let $\overline{S}_{k}\stackrel{\text{def}}{=}k[u]/u^{ep}$ .

Definition 2.3.1. Let ${\mathcal{M}}$ be an object in $\mathbb{F}\text{-}\operatorname{BrMod}_{\text{dd}}^{r}$ . An $\overline{S}$ -submodule ${\mathcal{N}}\subseteq {\mathcal{M}}$ is said to be a Breuil submodule if ${\mathcal{N}}$ fulfills the following conditions:

  1. (i) ${\mathcal{N}}$ is an $\overline{S}_{k}$ -direct summand of ${\mathcal{M}}$ ;

  2. (ii) $N({\mathcal{N}})\subseteq {\mathcal{N}}$ and $\widehat{g}({\mathcal{N}})\subseteq {\mathcal{N}}$ for all $g\in \operatorname{Gal}(K/K^{\prime })$ ;

  3. (iii) $\unicode[STIX]{x1D711}_{r}({\mathcal{N}}\,\cap \,\operatorname{Fil}^{r}{\mathcal{M}})\subseteq {\mathcal{N}}$ .

Lemma 2.3.2. Let ${\mathcal{M}}$ be an object in $\mathbb{F}\text{-}\operatorname{BrMod}_{\text{dd}}^{r}$ and let ${\mathcal{N}}\subseteq {\mathcal{M}}$ be a Breuil submodule. Then the $\overline{S}$ -modules ${\mathcal{N}}$ , ${\mathcal{M}}/{\mathcal{N}}$ with their induced structures are objects of $\mathbb{F}\text{-}\operatorname{BrMod}_{\text{dd}}^{r}$ and the sequence

$$\begin{eqnarray}0\rightarrow {\mathcal{N}}\rightarrow {\mathcal{M}}\rightarrow {\mathcal{M}}/{\mathcal{N}}\rightarrow 0\end{eqnarray}$$

is exact in $\mathbb{F}\text{-}\operatorname{BrMod}_{\text{dd}}^{r}$ . Conversely, given an exact sequence

$$\begin{eqnarray}0\rightarrow {\mathcal{M}}_{1}\stackrel{f}{\rightarrow }{\mathcal{M}}\rightarrow {\mathcal{M}}_{2}\rightarrow 0\end{eqnarray}$$

in $\mathbb{F}\text{-}\operatorname{BrMod}_{\text{dd}}^{r}$ , then $f({\mathcal{M}}_{1})\subseteq {\mathcal{M}}$ is a Breuil submodule.

An immediate consequence of Lemma 2.3.2 is that the notion of Breuil submodule is transitive.

Lemma 2.3.3. Let ${\mathcal{M}}$ be an object in $\mathbb{F}\text{-}\operatorname{BrMod}_{\text{dd}}^{r}$ .

  1. (i) If ${\mathcal{M}}_{1}\subseteq {\mathcal{M}}$ and ${\mathcal{M}}_{2}\subseteq {\mathcal{M}}_{1}$ are Breuil submodules, then so is ${\mathcal{M}}_{2}\subseteq {\mathcal{M}}$ .

  2. (ii) Let ${\mathcal{M}}_{1}$ , ${\mathcal{M}}_{2}$ be Breuil submodules of ${\mathcal{M}}$ , and assume that ${\mathcal{M}}_{2}\subseteq {\mathcal{M}}_{1}$ . Then ${\mathcal{M}}_{2}$ is a Breuil submodule of ${\mathcal{M}}_{1}$ , and the Breuil module structures on ${\mathcal{M}}_{2}$ inherited from ${\mathcal{M}}_{1}$ and ${\mathcal{M}}$ coincide. Similarly, ${\mathcal{M}}_{1}/{\mathcal{M}}_{2}\subseteq {\mathcal{M}}/{\mathcal{M}}_{2}$ is a Breuil submodule, and the Breuil module structures on