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Florian Herzig

Publications and source records attributed to Florian Herzig.

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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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

On the irreducibility of $p$-adic Banach principal series of $p$-adic reductive groups

Suppose that $G$ is the group of $F$-points of a connected reductive group over $F$, where $F/\mathbb{Q}_p$ is a finite extension. We study the (topological) irreducibility of principal series of $G$ on $p$-adic Banach spaces. For unitary inducing representations we obtain an optimal irreducibility criterion, and for $G = \mathrm{GL}_n(F)$ (as well as for arbitrary split groups under slightly stronger conditions) we obtain a variant of Schneider's conjecture [Sch06, Conjecture 2.5]. In general we reduce the irreducibility problem to smooth inducing representations and almost simple simply-connected $G$. Our methods include locally analytic representation theory, the bifunctor of Orlik--Strauch, translation functors, as well as new results on reducibility points of smooth parabolic inductions.

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arXiv2023

On the irreducibility of $p$-adic Banach principal series of $p$-adic $\mathrm{GL}_3$

We establish an optimal (topological) irreducibility criterion for $p$-adic Banach principal series of $\mathrm{GL}_{n}(F)$, where $F/\mathbb{Q}_p$ is finite and $n \le 3$. This is new for $n = 3$ as well as for $n = 2$, $F \ne \mathbb{Q}_p$ and establishes a refined version of Schneider's conjecture [Sch06, Conjecture 2.5] for these groups.

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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

Inverse Satake isomorphism and change of weight

Let $G$ be any connected reductive $p$-adic group. Let $K\subset G$ be any special parahoric subgroup and $V,V'$ be any two irreducible smooth $\overline {\mathbb F}_p[K]$-modules. The main goal of this article is to compute the image of the Hecke bi-module $\operatorname{End}_{\overline {\mathbb F}_p[K]}(\operatorname{c-Ind}_K^G V, \operatorname{c-Ind}_K^G V')$ by the generalized Satake transform and to give an explicit formula for its inverse, using the pro-$p$ Iwahori Hecke algebra of $G$. This immediately implies the "change of weight theorem" in the proof of the classification of mod $p$ irreducible admissible representations of $G$ in terms of supersingular ones. A simpler proof of the change of weight theorem, not using the pro-$p$ Iwahori Hecke algebra or the Lusztig-Kato formula, is given when $G$ is split (and in the appendix when $G$ is quasi-split, for almost all $K$).

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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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arXiv2019

Towards the finite slope part for $\mathrm{GL}_n$

Let $L$ be a finite extension of $\mathbb{Q}_p$ and $n\geq 2$. We associate to a crystabelline $n$-dimensional representation of $\mathrm{Gal}(\overline L/L)$ satisfying mild genericity assumptions a finite length locally $\mathbb{Q}_p$-analytic representation of $\mathrm{GL}_n(L)$. In the crystalline case and in a global context, using the recent results on the locally analytic socle from [BHS17a] we prove that this representation indeed occurs in spaces of $p$-adic automorphic forms. We then use this latter result in the ordinary case to show that certain "ordinary" $p$-adic Banach space representations constructed in our previous work appear in spaces of $p$-adic automorphic forms. This gives strong new evidence to our previous conjecture in the $p$-adic case.

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arXiv2017

General Serre weight conjectures

We formulate a number of related generalisations of the weight part of Serre's conjecture to the case of GL(n) over an arbitrary number field, motivated by the formalism of the Breuil-Mézard conjecture. We give evidence for these conjectures, and discuss their relationship to previous work. We generalise one of these conjectures to the case of connected reductive groups which are unramified over Q_p, and we also generalise the second author's previous conjecture for GL(n)/Q to this setting, and show that the two conjectures are generically in agreement.

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arXiv2017

On mod p local-global compatibility for GL_3 in the ordinary case

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 rbar : Gal(F-bar/F) -> GL_3(Fp-bar) is a continuous irreducible Galois representation such that rbar|_{Gal(F_w-bar/F_w)} is upper-triangular, maximally non-split, and generic. If rbar is automorphic, and some suitable technical conditions hold, we show that rbar|_{\Gal(F_w-bar/F_w)} can be recovered from the 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 rbar, show the existence of an ordinary lifting of rbar, and prove the freeness of certain Taylor-Wiles patched modules in this context. We also show the existence of many Galois representations rbar to which our main theorem applies.

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arXiv2017

Adequate subgroups and indecomposable modules

The notion of adequate subgroups was introduced by Jack Thorne [59]. It is a weakening of the notion of big subgroups used by Wiles and Taylor in proving automorphy lifting theorems for certain Galois representations. Using this idea, Thorne was able to strengthen many automorphy lifting theorems. It was shown in [22] and [23] that if the dimension is smaller than the characteristic then almost all absolutely irreducible representations are adequate. We extend the results by considering all absolutely irreducible modules in characteristic p of dimension p. This relies on a modified definition of adequacy, provided by Thorne in [60], which allows p to divide the dimension of the module. We prove adequacy for almost all irreducible representations of SL_2(p^a) in the natural characteristic and for finite groups of Lie type as long as the field of definition is sufficiently large. We also essentially classify indecomposable modules in characteristic p of dimension less than 2p-2 and answer a question of Serre concerning complete reducibility of subgroups in classical groups of low dimension.

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arXiv2017

Potentially crystalline lifts of certain prescribed types

We prove several results concerning the existence of potentially crystalline lifts with prescribed Hodge-Tate weights and inertial types of a given n-dimensional mod p representation of the absolute Galois group of K, where K/Q_p is a finite extension. Some of these results are proved by purely local methods, and are expected to be useful in the application of automorphy lifting theorems. The proofs of the other results are global, making use of automorphy lifting theorems.

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arXiv2016

A classification of irreducible admissible mod p representations of p-adic reductive groups

Let F be a locally compact non-archimedean field, p its residue characteristic, and G a connected reductive group over F. Let C an algebraically closed field of characteristic p. We give a complete classification of irreducible admissible C-representations of G = G(F), in terms of supercuspidal C-representations of the Levi subgroups of G, and parabolic induction. Thus we push to their natural conclusion the ideas of the third-named author, who treated the case G = GL_m, as further expanded by the first-named author, who treated split groups G. As in the split case, we first get a classification in terms of supersingular representations of Levi subgroups, and as a consequence show that supersingularity is the same as supercuspidality.

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arXiv2015

Ordinary representations of G(Q_p) and fundamental algebraic representations

Let G be a split connected reductive algebraic group over Q_p such that both G and its dual group G-hat have connected centres. Motivated by a hypothetical p-adic Langlands correspondence for G(Q_p) we associate to an n-dimensional ordinary (i.e. Borel valued) representation rho : Gal(Q_p-bar/Q_p) to G-hat(E) a unitary Banach space representation Pi(rho)^ord of G(Q_p) over E that is built out of principal series representations. (Here, E is a finite extension of Q_p.) Our construction is inspired by the "ordinary part" of the tensor product of all fundamental algebraic representations of G. There is an analogous construction over a finite extension of F_p. In the latter case, when G=GL_n we show under suitable hypotheses that Pi(rho)^ord occurs in the rho-part of the cohomology of a compact unitary group. We also prove a weaker version of this result in the p-adic case.

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