arXiv subjects

Stefano Morra

Publications and source records attributed to Stefano Morra.

At least 19 recordsLinked to original sources

arXiv2026

To be or not to be local

Let $p$ be a prime number and $K$ a finite unramified extension of $\mathbf{Q}_p$. For a smooth representation $\pi$ of $\mathrm{GL}_2(K)$ occurring in some Hecke eigenspace of the mod $p$ cohomology of a Shimura curve, we explore different strategies (inspired by the case $K=\mathbf{Q}_p$) to attack the locality question: does $\pi$ depend only on the underlying $2$-dimensional representation $\overline{\rho}$ of ${\rm Gal}(\overline K/K)$? In particular when $[K:\mathbf{Q}_p]=2$, crucially using perfectoid geometry, we associate to $\overline{\rho}$ an infinite-dimensional mod $p$ smooth representation of $\begin{pmatrix}K^\times&K\\0&1\end{pmatrix}$ which we hope is the restriction to $\begin{pmatrix}K^\times&K\\0&1\end{pmatrix}$ of the (irreducible) supersingular subquotient of $\pi$.

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arXiv2025

On the constituents of the mod $p$ cohomology of Shimura curves

Let $p$ be a prime number and $K$ a finite unramified extension of $\mathbb{Q}_p$. When $p$ is large enough with respect to $[K:\mathbb{Q}_p]$ and under mild genericity assumptions, we proved in our previous work that the admissible smooth representations $\pi$ of $\mathrm{GL}_2(K)$ that occur in Hecke eigenspaces of the mod $p$ cohomology are of finite length. In this paper we obtain various refined results about the structure of subquotients of $\pi$, such as their Iwahori-socle filtrations and $K_1$-invariants, where $K_1$ is the principal congruence subgroup of $\mathrm{GL}_2(\mathcal{O}_K)$. We also determine the Hilbert series of $\pi$ as Iwahori-representation under these conditions.

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arXiv2025

Finite length for unramified $\mathrm{GL}_2$

Let $p$ be a prime number and $K$ a finite unramified extension of $\mathbb{Q}_p$. If $p$ is large enough with respect to $[K:\mathbb{Q}_p]$ and under mild genericity assumptions, we prove that the admissible smooth representations of $\mathrm{GL}_2(K)$ that occur in Hecke eigenspaces of the mod $p$ cohomology are of finite length. We also prove many new structural results about these representations of $\mathrm{GL}_2(K)$ and their subquotients.

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arXiv2024

Generic decompositions of Deligne--Lusztig representations

Let $G_0$ be a reductive group over $\mathbb{F}_p$ with simply connected derived subgroup, (geometrically) connected center and Coxeter number $h+1$. We extend Jantzen's generic decomposition pattern from $(2h-1)$-generic to $h$-generic Deligne--Lusztig representations, which is optimal. We also prove several results on the ``obvious'' Jordan--H\"older factors of general Deligne--Lusztig representations. As an application we improve the weight elimination result of arXiv:1610.04819 [math.NT]

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arXiv2024

$K_1$-invariants in the mod $p$ cohomology of $U(3)$ arithmetic manifolds

Let $F/F^+$ be a CM extension and $H_{/F^+}$ a definite unitary group in three variables that splits over $F$. We describe Hecke isotypic components of mod $p$ algebraic modular forms on $H$ at first principal congruence level at $p$ and "minimal" level away from $p$ in terms of the restrictions of the associated Galois representation to decomposition groups at $p$ when these restrictions are tame and sufficiently generic. This confirms an expectation of local-global compatibility in the mod $p$ Langlands program. To prove our result, we develop a local model theory for multitype deformation rings and new methods to work with patched modules that are not free over their scheme-theoretic support.

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arXiv2024

Local model theory for non-generic tame potentially Barsotti--Tate deformation rings

We develop a local model theory for moduli stacks of $2$-dimensional non-scalar tame potentially Barsotti--Tate Galois representations of the Galois group of an unramified extension of $\mathbb{Q}_p$. We derive from this explicit presentations of potentially Barsotti--Tate deformation rings, allowing us to prove structural results about them, and prove various conjectures formulated by Caruso--David--Mézard.

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arXiv2023

Moduli of Fontaine--Laffaille representations and a mod-$p$ local-global compatibility result

Let $F/F^+$ be a CM field and let $\widetilde{v}$ be a finite unramified place of $F$ above the prime $p$. Let $\overline{r}: \mathrm{Gal}(\overline{\mathbb{Q}}/F)\rightarrow \mathrm{GL}_n(\overline{\mathbb{F}}_p)$ be a continuous representation which we assume to be modular for a unitary group over $F^+$ which is compact at all real places. We prove, under Taylor--Wiles hypotheses, that the smooth $\mathrm{GL}_n(F_{\widetilde{v}})$-action on the corresponding Hecke isotypical part of the mod-$p$ cohomology with infinite level above $\widetilde{v}|_{F^+}$ determines $\overline{r}|_{\mathrm{Gal}(\overline{\mathbb{Q}}_p/F_{\widetilde{v}})}$, when this latter restriction is Fontaine--Laffaille and has a suitably generic semisimplification.

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arXiv2023

Gelfand-Kirillov dimension for mod $p$ representations of $p$-adic unitary groups of rank 2

Let $p$ be a prime number and $F/F^+$ a CM extension of a totally real field such that every place of $F^+$ above $p$ is unramified and inert in $F$. We fix a finite place $v$ of $F^+$ above $p$, and let $\overline{r}: \textrm{Gal}(\overline{F^+}/F^+) \longrightarrow {}^C\textrm{U}_{1,1}(\overline{\mathbb{F}}_p)$ be a modular $L$-parameter valued in the $C$-group of a rank 2 unitary group associated to $F/F^+$. We assume $\overline{r}$ is semisimple and sufficiently generic at $v$. Using recent results of Breuil--Herzig--Hu--Morra--Schraen along with our previous work, we prove that certain admissible smooth $\overline{\mathbb{F}}_p$-representations of the $p$-adic unitary group $\textrm{U}_{1,1}(F^+_v)$ associated to $\overline{r}$ in spaces of mod $p$ automorphic forms have Gelfand--Kirillov dimension $[F^+_v:\mathbb{Q}_p]$.

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arXiv2023

Gelfand-Kirillov dimension and mod p cohomology for GL2

Let $p$ be a prime number, $F$ a totally real number field unramified at places above $p$ and $D$ a quaternion algebra of center $F$ split at places above $p$ and at no more than one infinite place. Let $v$ be a fixed place of $F$ above $p$ and $\overline{r} : {\rm Gal}(\overline F/F)\rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ an irreducible modular continuous Galois representation which, at the place $v$, is semisimple and sufficiently generic (and satisfies some weak genericity conditions at a few other finite places). We prove that many of the admissible smooth representations of $\mathrm{GL}_2(F_v)$ over $\overline{\mathbb{F}}_p$ associated to $\overline{r}$ in the corresponding Hecke-eigenspaces of the mod $p$ cohomology have Gelfand--Kirillov dimension $[F_v:\mathbb{Q}]$, as well as several related results.

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arXiv2023

Colength one deformation rings

Let $K/\mathbf{Q}_p$ be a finite unramified extension, $\overlineρ:\mathrm{Gal}(\overline{\mathbf{Q}}_p/K)\rightarrow\mathrm{GL}_n(\overline{\mathbf{F}}_p)$ a continuous representation, and $τ$ a tame inertial type of dimension $n$. We explicitly determine, under mild regularity conditions on $τ$, the potentially crystalline deformation ring $R^{η,τ}_{\overlineρ}$ in parallel Hodge--Tate weights $η=(n-1,\cdots,1,0)$ and inertial type $τ$ when the \emph{shape} of $\overlineρ$ with respect to $τ$ has colength at most one. This has application to the modularity of a class of shadow weights in the weight part of Serre's conjecture. Along the way we make unconditional the local-global compatibility results of \cite{PQ} and further study the geometry of moduli spaces of Fontaine--Laffaille representations in terms of colength one weights.

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arXiv2022

Multivariable ($\varphi$,$\mathcal{O}_K^\times$)-modules and local-global compatibility

Let $p$ be a prime number, $K$ a finite unramified extension of $\mathbb{Q}_p$ and $\mathbb{F}$ a finite extension of $\mathbb{F}_p$. Using perfectoid spaces we associate to any finite-dimensional continuous representation $\overline{\rho}$ of ${\rm Gal}(\overline K/K)$ over $\mathbb{F}$ an \'etale $(\varphi,\mathcal{O}_K^\times)$-module $D_A^\otimes(\overline{\rho})$ over a completed localization $A$ of $\mathbb{F}[\![\mathcal{O}_K]\!]$. We conjecture that one can also associate an \'etale $(\varphi,\mathcal{O}_K^\times)$-module $D_A(\pi)$ to any smooth representation $\pi$ of $\mathrm{GL}_2(K)$ occurring in some Hecke eigenspace of the mod $p$ cohomology of a Shimura curve, and that moreover $D_A(\pi)$ is isomorphic (up to twist) to $D_A^\otimes(\overline{\rho})$, where $\overline{\rho}$ is the underlying $2$-dimensional representation of ${\rm Gal}(\overline K/K)$. Using previous work of the same authors, we prove this conjecture when $\overline{\rho}$ is semi-simple and sufficiently generic.

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arXiv2022

Local models for Galois deformation rings and applications

We construct projective varieties in mixed characteristic whose singularities model, in generic cases, those of tamely potentially crystalline Galois deformation rings for unramified extensions of $\mathbb{Q}_p$ with small regular Hodge-Tate weights. We establish several significant facts about their geometry including a unibranch property at special points and a representation theoretic description of the irreducible components of their special fibers. We derive from these geometric results a number of local and global consequences: the Breuil-Mézard conjecture in arbitrary dimension for tamely potentially crystalline deformation rings with small Hodge-Tate weights (with appropriate genericity conditions), the weight part of Serre's conjecture for $U(n)$ as formulated by Herzig (for global Galois representations which satisfy the Taylor-Wiles hypotheses and are sufficiently generic at $p$), and an unconditional formulation of the weight part of Serre's conjecture for wildly ramified representations.

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arXiv2022

Extremal weights and a tameness criterion for mod $p$ Galois representations

We study the weight part of Serre's conjecture for generic $n$-dimensional mod $p$ Galois representations. We first generalize Herzig's conjecture to the case where the field is ramified at $p$ and prove the weight elimination direction of our conjecture. We then introduce a new class of weights associated to $n$-dimensional local mod $p$ representations which we call \emph{extremal weights}. Using a ``Levi reduction" property of certain potentially crystalline Galois deformation spaces, we prove the modularity of these weights. As a consequence, we deduce the weight part of Serre's conjecture for unit groups of some division algebras in generic situations.

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arXiv2022

Serre weights for three-dimensional wildly ramified Galois representations

We formulate and prove the weight part of Serre's conjecture for three-dimensional mod $p$ Galois representations under a genericity condition when the field is unramified at $p$. This removes the assumption in \cite{arXiv:1512.06380}, \cite{arXiv:1608.06570} that the representation be tamely ramified at $p$. We also prove a version of Breuil's lattice conjecture and a mod $p$ multiplicity one result for the cohomology of $U(3)$-arithmetic manifolds. The key input is a study of the geometry of the Emerton--Gee stacks \cite{arXiv:2012.12719} using the local models introduced in \cite{arXiv:2007.05398}.

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arXiv2021

Serre weight conjectures for $p$-adic unitary groups of rank 2

We prove a version of the weight part of Serre's conjecture for mod $p$ Galois representations attached to automorphic forms on rank 2 unitary groups which are non-split at $p$. More precisely, let $F/F^+$ denote a CM extension of a totally real field such that every place of $F^+$ above $p$ is unramified and inert in $F$, and let $\overline{r}: \textrm{Gal}(\overline{F^+}/F^+) \longrightarrow {}^C\mathbf{U}_2(\overline{\mathbb{F}}_p)$ be a Galois parameter valued in the $C$-group of a rank 2 unitary group attached to $F/F^+$. We assume that $\overline{r}$ is semisimple and sufficiently generic at all places above $p$. Using base change techniques and (a strengthened version of) the Taylor-Wiles-Kisin conditions, we prove that the set of Serre weights in which $\overline{r}$ is modular agrees with the set of Serre weights predicted by Gee-Herzig-Savitt.

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arXiv2021

Conjectures and results on modular representations of $\mathrm{GL}_n(K)$ for a $p$-adic field $K$

Let $p$ be a prime number and $K$ a finite extension of $\mathbb{Q}_p$. We state conjectures on the smooth representations of $\mathrm{GL}_n(K)$ that occur in spaces of mod $p$ automorphic forms (for compact unitary groups). In particular, when $K$ is unramified, we conjecture that they are of finite length and predict their internal structure (extensions, form of subquotients) from the structure of a certain algebraic representation of $\mathrm{GL}_n$. When $n=2$ and $K$ is unramified, we prove several cases of our conjectures, including new finite length results.

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arXiv2020

Multiplicity one at full congruence level

Let $F$ be a totally real field in which $p$ is unramified. Let $\overline{r}: G_F \rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a modular Galois representation which satisfies the Taylor--Wiles hypotheses and is tamely ramified and generic at a place $v$ above $p$. Let $\mathfrak{m}$ be the corresponding Hecke eigensystem. We describe the $\mathfrak{m}$-torsion in the mod $p$ cohomology of Shimura curves with full congruence level at $v$ as a $\mathrm{GL}_2(k_v)$-representation. In particular, it only depends on $\overline{r}|_{I_{F_v}}$ and its Jordan--Hölder factors appear with multiplicity one. The main ingredients are a description of the submodule structure for generic $\mathrm{GL}_2(\mathbb{F}_q)$-projective envelopes and the multiplicity one results of \cite{EGS}.

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arXiv2020

Serre weights and Breuil's lattice conjecture in dimension three

We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a $U(3)$-arithmetic manifold is purely local, i.e., only depends on the Galois representation at places above $p$. This is a generalization to $\mathrm{GL}_3$ of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil-Mézard conjecture for (tamely) potentially crystalline deformation rings with Hodge-Tate weights $(0,1,2)$ as well as the Serre weight conjectures over an unramified field extending our previous results. We also prove results in modular representation theory about lattices in Deligne-Luzstig representations for the group $\mathrm{GL}_3(\mathbb{F}_q)$.

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