Bonn-HIM series in 2023 (II): the Arithmetic of the Langlands Program. Supported in part by the Philip Leverhulme Prize of Ana Caraiani.
This conference will be on various aspects of the local Langlands correspondence over p-adic fields and methods from p-adic Hodge theory. Topics will include the usual local Langlands correspondence, the p-adic local Langlands correspondence and the relation to coherent sheaves on spaces of Galois representations, and the geometry and cohomology of local Shimura varieties.
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Tasho Kaletha (University of Michigan) – A local Langlands conjecture for disconnected groups. (notes)
Langlands’ conjectures are usually phrased in the setting of connected reductive groups. In this talk we will explore a generalization of the statement of the local conjectures to the setting of disconnected groups, subject to a certain mild restriction. Proving these conjectures in the simplest case – that of a disconnected group whose identity component is a torus – already shows an interesting new phenomenon. Time permitting, we will mention an application to the reduction of the refined local Langlands correspondence, both for connected and disconnected groups, to the case of discrete parameters.
Charlotte Chan (University of Michigan) – Generic character sheaves for parahoric subgroups. (notes)
Lusztig’s theory of character sheaves for connected reductive groups is one of the most important developments in representation theory in the last few decades. In this talk, we will describe a construction which extends this “depth zero” picture to give positive-depth character sheaves associated to generic data. In the simplest nontrivial case, this resolves a conjecture of Lusztig and produces perverse sheaves associated to suffciently generic multiplicative local systems whose trace-of-Frobenius function coincides with parahoric Deligne-Lusztig induction. This is joint work with R. Bezrukavnikov.
Andrea Dotto (University of Chicago) – Some consequences of mod p multiplicity one for Shimura curves. (notes)
The multiplicity of Hecke eigenspaces in the mod p cohomology of Shimura curves is a classical invariant, which has been computed in significant generality when the group is split at p. This talk will focus on the complementary case of nonsplit quaternion algebras, and will describe a new multiplicity one result, as well as some of its consequences regarding the structure of completed cohomology. The speaker will also discuss applications towards the categorical mod p Langlands correspondence for the nonsplit inner form of GL(2,Qp). Part of the talk will comprise a joint work in progress with Bao Le Hung.
David Helm (Imperial College London) – Finiteness for Hecke algebras of p-adic reductive groups. (notes)
Let F be a p-adic field and $G$ a connected reductive group over $F$. It is a classical theorem of Bernstein that for any compact open subgroup $U$ of $G(F)$, the Hecke algebra $C[U/G(F)/U]$ is Noetherian; in fact its center is a finitely generated $C$-algebra and the Hecke algebra itself is a finitely generated module over its center. A longstanding and surprisingly dificult problem has been to establish analogous results over coeficient rings other than $C$, such as $\ell$-adic integer rings for $\ell$ different from p. We explain how the recent results of Fargues-Scholze towards a categorical local Langlands correspondence provide the missing ingredient to resolve this problem.
Jessica Fintzen (Universität Bonn) – Representations of p-adic groups and Hecke algebras. (notes)
An explicit understanding of the category of all (smooth, complex) representations of p-adic groups provides an important tool in the construction of an explicit or categorical local Langlands correspondence. The category of representations of p-adic groups decomposes into subcategories, called Bernstein blocks, which are indexed by equivalence classes of so called supercuspidal representations of Levi subgroups. In this talk, the speaker will give an overview of what we know about an explicit construction of supercuspidal representations and about the structure of the Bernstein blocks. In particular, The speaker will discuss a joint project in progress with Jeffrey Adler, Manish Mishra and Kazuma Ohara in which we expect to show that general Bernstein blocks are equivalent to much better understood depth-zero Bernstein blocks. This is achieved via an isomorphism of Hecke algebras.
Eugen Hellmann (Universität Münster) – The finite slope part of overconvergent cohomology and coherent sheaves on spaces of Galois representations. (notes)
Overconvergent cohomology of Shimura varieties is a well studied object on the context of (overconvergent) p-adic automorphic forms. The construction of eigenvarieties gives rise to a coherent sheaf on a rigid analytic space whose cohomology can be identified with (the dual of) the finite slope part of the overconvergent cohomology. The speaker will state a general conjecture how to compute this coherent sheaf in terms of stacks of Galois representations and explain how this conjecture fits into the emerging categorical approach to the p-adic Langlands program. The speaker will also explain joint work in progress with Zhixiang Wu in which we prove the conjecture in the case of the modular curve(s).
Juan Esteban Rodriguez Camargo (MPIM Bonn) – A local Jacquet-Langlands correspondence for locally analytic D-modules. (notes)
In this talk the speaker will explain an equivalence between the categories of locally analytic $\mathrm{GL}(2,F)$-equivariant D-modules over the Drinfeld space, and the locally analytic $D^{\ast}$-equivariant D-modules over $\mathbb{P}^1$ from the Lubin-Tate side. As an application, we prove that the locally analytic version of the functor of Scholze preserves admissible locally analytic representations. This is joint work in progress with Gabriel Dospinescu.
Eva Viehmann (Universität Münster) – Admissibility and weak admissibility. (notes)
The speaker will give an overview of the relation between the weakly admissible locus and the admis- sible locus, and of the associated Harder-Narasimhan stratification resp. Newton stratification of the de Rham $B^+$-affine Grassmannian.
David Hansen (National University of Singapore) – What can categorical local Langlands do for YOU?. (notes)
The speaker will formulate some conjectures relating the categorical and classical local Langlands correspondences, and discuss what is known about them.
Gabriel Dospinescu (UMPA ENS de Lyon) – One-dimensional mod p quaternionic representations. (notes)
The speaker will explain how the recent six functor formalism of Lucas Mann together with finiteness results of Scholze imply some rather concrete vanishing and finiteness properties for the mod _p cohomology of the Lubin-Tate and Drinfeld spaces in dimension 1, and related results for Hecke eigenspaces in mod p cohomology of quaternionic Shimura curves ramified at p. This is joint work (in progress) with Juan Esteban Rodríguez Camargo._
Wieslawa Niziol (CNRS/Sorbonne University) – Duality in p-adic pro-etale cohomology of analytic curves. (notes)
The speaker will discuss duality theorems in p-adic pro-etale cohomology of analytic curves. This is joint work with Pierre Colmez and Sally Gilles.
Ian Gleason (Universität Bonn) – Meromorphic vector bundles on the Fargues-Fontaine curve. (notes)
Recent progress in p-adic and perfect geometry has led experts to conjecture a precise categorical (or geometric) version of the local Langlands correspondence. There are mainly two approaches to state the conjecture, one using p-adic geometry and the stack of vector bundles on the Fargues-Fontaine curve and the other one using perfect geometry and the Kottwitz stack parametrizing families of isocrystals. One can ask how do the two conjectures compare. Motivated by this question we introduce and study a third object, the stack of meromorphic vector bundles on the Fargues-Fontaine curve. We provide evidence that this object should “mediate” between the two conjectures. This is joint work in progress with Alex Ivanov.
João Lourenço (WWU Münster) – Towards Bezrukavnikov via p-adic central sheaves. (notes)
In the 80s Kazhdan-Lusztig computed the equivariant coherent Grothendieck group of the Steinberg variety, which identifies with the equivariant étale Grothendieck group of the corresponding affine flag variety in equicharacteristic. A decade ago, Bezrukavnikov upgraded this isomorphism to an equivalence of derived categories of sheaves, building on the central sheaves constructed by Gaitsgory and his work with Arkhipov on the anti-spherical realization. Joint with J. Anschütz, I. Gleason, and T. Richarz, we constructed central sheaves inside the Witt vector affne flag variety in terms of perfectoid nearby cycles. We will explain joint progress with J. Anschütz, Z. Wu, and J. Yu towards writing down an Arkhipov-Bezrukavnikov equivalence building on those p-adic central sheaves.
Marie-France Vigneras (Université de Paris) – Representations of GL(n,D) near the identity. (notes)
For mod p irreducible admissible representations $\pi$ of $G = \mathrm{GL}(2,\mathbb{Q}p)$, parahoric subgroups $\mathfrak{k}^{\ast}$ of $G$ of congruence subgroups $K(j) = 1+p^j \mathfrak{k}$, $j \geqslant 1$, Morra (2013) proved that $\operatorname{dim} \pi^{K(j)}=a+bp^j$ with two explicit integers $a, b$. The speaker will explain that a similar result holds true for the representations of the inner forms of the general linear groups over a finite extension of $\mathbb{Q}p$ or of $\mathbb{F}p((t))$, when the coefficient field has characteristic $\neq p$. This is common work with Guy Henniart (2023).
Lucas Mann (Universität Münster) – Representation Theory via 6-Functor Formalisms. (notes)
We present recent advances on the abstract theory of 6-functor formalisms and apply them to the representation theory of locally profinite groups. This sheds new light on classical results like preservation of admissibility under various operations, Bernstein-Zelevinsky duality and Second Adjointness. As an application we obtain new results on the p-adic representation theory of p-adic Lie groups.
Ashwin Iyengar (Johns Hopkins University) – Gamma factors for mod $\ell$ representations of finite general linear groups. (notes)
In the spirit of works going back to Jacquet-Piatetski-Shapiro-Shalika’s “Rankin-Selberg convolutions” paper, we construct gamma factors for mod $\ell$ representations of finite general linear groups and use them to prove a converse theorem in this context. The speaker will explain how the converse theorem fails for a naïve notion of gamma factor, and then explain how we can get around this failure by enlarging the rings of coefficients in which our upgraded gamma factors take values. The speaker will also discuss the relationship to recent work of Li-Shotton which addresses a finite field analogue of “local Langlands in families”. This is joint work with J. Bakeberg, M. Gerbelli-Gauthier, G. Moss, H. Goodson, and R. Zhang.
Brandon Levin (Rice University) – The weight part of Serre’s conjecture and the Emerton-Gee stack. (notes)
The Breuil-Mezard conjecture predicts the geometry of local deformation rings with p-adic Hodge theory conditions in terms of modular representation theory. The speaker will describe a version of this conjecture on the Emerton-Gee moduli stack of mod p Galois representations and its connection with the weight part of Serre’s conjecture. The speaker will then discuss joint work with Daniel Le, Bao V. Le Hung, and Stefano Morra on both these conjectures for certain classes of potentially crystalline substacks.
Tony Feng (Berkeley) – A new approach to Breuil-Mezard cycles. (notes)
The Breuil-Mezard Conjecture predicts the existence of hypothetical “Breuil-Mezard cycles” in the moduli space of mod p Galois representations of Qp that should govern congruences between mod p automorphic forms on a reductive group $G$. For $G = \mathrm{GL}(2)$, it is closely related to the weight part of Serre’s Conjectures. Thus far the most general progress on the Breuil-Mezard Conjecture has been based on patching, which however has some limitations of global nature. The speaker will talk about joint work with Bao Le Hung on a new approach to the Breuil-Mezard Conjecture for generic parameters. Our method is purely local and group-theoretic; instead of patching, it is based on geometric representation theory and microlocal analysis. In particular, we connect the Breuil-Mezard Conjecture to questions concerning quantum groups and homological mirror symmetry.