# Trimester Seminar Series

**Venue:** HIM lecture hall, Poppelsdorfer Allee 45, Bonn

Organizers: Tony Feng, Kalyani Kansal, Arthur-César Le Bras, and David Schwein

**August 17, 2023 (CEST)**

3 - 4pm Teruhisa Koshikawa

Title: Vanishing range beyond the generic case

Abstract: I will talk about some ideas about the vanishing range of the cohomology with rational coefficients of Shimura varieties and locally symmetric spaces. Our aim is to understand the situation beyond the generic case. This is a joint project with Sug Woo Shin.

**August 14, 2023 (CEST)**

2 - 4pm Juan Esteban Rodriguez Camargo

Title: An introduction to geometric Sen theory

Abstract: In this talk I will explain the construction of the geometric Sen operator on rigid spaces and discuss different applications to Shimura varieties. More precisely, I will briefly recall basic facts on the theory of solid locally analytic representations, establish the set up for geometric Sen theory, explain the relation of the Sen operator with proétale cohomology, and show how to compute the Sen operator of a perfectoid Shimura variety.

**August 3, 2023 (CEST)**

3 – 4pm Mingjia Zhang

Title: Igusa stacks and p-adic Shimura varieties

Abstract: Scholze has conjectured that a p-adic Shimura variety as a diamond can be expressed as a fiber product of a flag variety with a certain small v-stack, over the stack Bun_G due to Fargues-Scholze. In line with this conjecture, I constructed small v-stacks ("Igusa stacks") for some PEL type Shimura varieties, which, together with the flag varieties uniformize the Shimura varieties in the desired way. I will explain the conjecture, the construction of the Igusa stacks and some applications.

**July 27, 2023 (CEST)**

3 – 4pm Konrad Zou

Title: Categorical local Langlands for GL_n: the irreducible case with

integral coefficients

Abstract: Fargues and Scholze conjecture a Hecke-equivariant equivalence of categories between certain coherent sheaves on the stack of Langlands parameters and compact objects in the category of lisse-etale sheaves on Bun_G. We will discuss how to prove this conjecture for irreducible parameters for GL_n, even with integral coefficients. It turns out that this needs surprisingly little knowledge about the spaces involved, the non-formal input is the cardinality of the Fargues-Scholze L-packets and genericity of their members. The formal input is about localizations of categories over schemes, which we will discuss.

**July 27, 2023 (CEST)**

1:30 –2:30pm Chenji Fu

Title: Explicit mod-\ell categorical local Langlands correspondence for

depth-zero supercuspidal part of GL_2

Abstract: Let F be a non-archimedean local field. I will explicitly describe:

(1) (the category of quasicoherent sheaves on) the connected component of the moduli space of Langlands parameters over Z_l-bar containing an irreducible tame L-parameter with F_l-bar coefficients;

(2) the block of the category of smooth representations of G(F) with Z_l-bar coefficients containing a depth-zero supercuspidal representation with F_l-bar coefficients.

The argument works at least for (simply connected) split reductive group G, but I will focus on the example of GL_2 for simplicity. The two sides turn out to match abstractly. If time permits, I will explain how to get the categorical local Langlands correspondence for depth-zero supercuspidal part of GL_2 with Z_l-bar coefficients in Fargues-Scholze's form.

I will try to upload my manuscript as well as handwritten notes on my github page: https://github.com/Chenji-Fu/Master-thesis

And you are welcome to send any comments to my email: carlfuchenji@gmail.com

July 20, 2023 (CEST)

3:00 - 4:00pm Peter Scholze

Title: Some remarks on prismatic cohomology of rigid spaces

Abstract: n.a.

July 20, 2023 (CEST)

1:30 - 2:30pm Quentin Gazda

Title: Motivic cohomology of Carlitz twists and its relation to zeta

values and polylogarithms

Abstract: Drinfeld modules, mostly understood as analogues of elliptic curves, are central objects of the arithmetic of function fields. The study of their theory of moduli via the fundamental notion of shtukas led Drinfeld, and then L. and V. Lafforgue, to fundamental achievements in the Langlands correspondence for reductive groups over function fields. It was also realized that Drinfeld modules are interesting objects of number theory by themselves : their theory is remarkably similar to that of abelian varieties, with analogs of zeta values, Tate modules and transcendental periods. This led Anderson to interpret a slight variant of the category of Drinfeld’s shtukas as playing the role of a category of motives. Following the close analogy among Anderson’s t-motives and classical motives, I will introduce t-motivic cohomology (the counterpart of motivic cohomology in this setting). I will mostly survey on recent computations in the simplest case of the Carlitz twists — the counterpart of Tate twists — and their relations the function fields zeta values and polylogarithms (joint with A. Maurischat).

July 13, 2023 (CEST)

3:00 - 4:00pm Alexander Petrov

Title: On de Rham cohomology in positive characteristic

Abstract: Deligne and Illusie established an analog of Hodge decomposition in positive characteristic: for a smooth proper variety X over F_p equipped with a lift over Z/p^2, there is a natural isomorphism between de Rham and Hodge cohomology, provided that the dimension of X is at most p. It turns out that the analogous isomorphism might fail for liftable varieties of dimension larger than p (that is, the de Rham cohomology might have smaller dimension than Hodge cohomology). This failure can be seen as coming from the non-vanishing of the cohomology of reductive groups in positive characteristics combined with the different behaviour of Steenrod operations on de Rham and Hodge cohomology. I will also discuss some structures that are nonetheless always present on the de Rham complex of variety over F_p, such as the Sen operator of Drinfeld and Bhatt-Lurie in the presence of a lift over Z/p^2, and the canonical decomposition after the Frobenius pullback.

July 6, 2023 (CEST)

3:00 - 4:00pm Sally Gilles

Title: On compactly supported p-adic proétale cohomology of analytic

varieties.

Abstract: I will define the p-adic proétale cohomology with compact support for analytic varieties and present some properties that it satisfies. In particular, I will discuss comparison theorems between the (compactly supported versions of) proétale and de Rham cohomologies.This is a joint work with Piotr Achinger and Wieslawa Niziol.

July 6, 2023 (CEST)

1:30 - 2:30pm Sean Howe

Title: Differential topology for diamonds

Abstract: In this talk, I will construct explicit Banach-Colmez tangent spaces for many diamonds that arise naturally in the study of smooth rigid analytic varieties and their cohomology. In an ideal world, these tangent spaces would be defined in terms of some theory of analytic structures on diamonds, but I do not have any such theory to propose! Nevertheless, starting from first principles, I will explain* why the tangent spaces constructed must be the correct ones if any such theory exists. Once we have the tangent spaces, it is natural to make some conjectures in the spirit of differential topology describing underlying properties of the diamond in terms of this extra differential data: in particular, we will formulate some conjectures of this nature describing preperfectoid loci and cohomological smoothness of morphisms. Along the way we will give explicit examples and computations of tangent spaces and derivatives in this context and compare our conjectures with known results and other related work.

* This explanation will inevitably involve a short but compelling technical jaunt through geometric Sen theory and the p-adic Simpson and Riemann-Hilbert correspondence targeted at actual or hypothetical experts who may or may not be in the room. You may or may not find this jaunt independently interesting, but you are hereby and henceforth expressly permitted to blackbox everything or even just ignore it completely.

June 29, 2023 (CEST)

3:00 - 4:00pm Thibaud van den Hove

Title: The integral motivic Satake equivalence

Abstract: For a reductive group G over a field k, geometric Satake gives an equivalence between the category of equivariant perverse sheaves on the affine Grassmannian of G and the category of representations of the Langlands dual group of G. Depending on the field k, one can use different cohomology theories, such as Betti cohomology, étale cohomology, (arithmetic) D-modules, ... On the other hand, the representation category of the Langlands dual group remains the same, depending only on the coefficients of the cohomology theory. In this talk, I will explain how to construct a version of the Satake equivalence using a universal cohomology theory, i.e., motivically, and with integral coefficients. This generalizes and unifies many previously known instances of geometric Satake. This is joint work with Robert Cass and Jakob Scholbach.

June 29, 2023 (CEST)

1:30 - 2:30pm Kalyani Kansal

Title: Non-regular loci in the Emerton-Gee stack for GL2

**Abstract:** Let K be a finite extension of Qp. The Emerton-Gee stack for GL2 is a stack of etale (phi, Gamma)-modules of rank two. Its reduced part, X, is an algebraic stack of finite type over a finite field, and can be viewed as a moduli stack of two dimensional mod p representations of the absolute Galois group of K. By the work of Caraiani, Emerton, Gee and Savitt, it is known that except for Steinberg components, the loci of mod p representations admitting crystalline lifts with small, regular Hodge-Tate weights are precisely the irreducible components of X. Their work relied on a detailed study of a closely related stack of etale phi-modules which admits a map from a stack of Breuil-Kisin modules with descent data. In our work, we assume K is unramfied and further study this map with a view to studying the loci of mod p representations admitting crystalline lifts with small, non-regular Hodge-Tate weights. We identify these loci as images of certain irreducible components of the stack of Breuil-Kisin modules and produce a "shape-shifting" algorithm to obtain several inclusions of the non-regular loci into the irreducible components of X. This is joint work in progress with Rebecca Bellovin, Neelima Borade, Anton Hilado, Heejong Lee, Brandon Levin, David Savitt and Hanneke Wiersema.

June 22, 2023 (CEST)

3:00 - 4:00pm Eugen Hellmann

Title: Explicit examples of categorical local Langlands

Abstract: Categorical local Langlands aims to establish a relation between the derived category of smooth representations of a p-adic reductive group and the derived category of coherent sheaves on the stack of corresponding L-parameters. In this talk we will discuss some examples, mainly in the case of the groups GL_2 and SL_2, where one can construct an explicit fully faithful functor between these categories. In these cases we will explicitly compute the (complexes of) sheaves associated to certain smooth representations and discuss how parabolic induction, the decomposition into Bernstein blocks and the Langlands classification of irreducible representations fits into this picture.

June 22, 2023 (CEST)

1:30 - 2:30pm Valentin Hernandez

Title: On classicality of p-adic modular forms

Abstract: p-adic modular forms are an essential object of study in arithmetic geometry, but it is also important to know when a p-adic modular form is actually a classical modular forms.

For GL_2, on the Eigencurve, theorems of Coleman and Kisin assures that a p-adic form is classical if its weight is positive, and the slope is small or if the Galois representation crystalline. The Fontaine-Mazur conjecture assures that if the Galois representation of a p-adic modular form is crystalline at p, it should be associated to a classical modular form, but possibly a different one. Already for GL_2 examples proves that crystalline p-adic modular forms need not to be classical if their weight is not « classical » or if they are not Hecke eigenvector at p. We could expect that a p-adic modular form which is an Eigenvector for all Hecke operators, of classical weight, and crystalline at p is classical. We show that this is not necessarily the case already for GL_3. This is a joint work with Eugen Hellmann and Benjamin Schraen.

June 8, 2023 (CEST)

3:00 - 4:00pm Daniel Li-Huerta (Harvard)

Title: Local-global compability over funtion fields

Abstract: We present a proof that V. Lafforgue's global Langlands correspondence is compatible with Fargues–Scholze's semisimplified local Langlands correspondence. By globalizing representations, this has the following local consequences:

- Fargues–Scholze's construction canonically lifts to a non-semisimplified correspondence in characteristic ≥ 5,
- Genestier–Lafforgue's correspondence agrees with Fargues–Scholze's.

The proof relies on a formal model for the moduli of local shtukas with multiple legs.

June 1, 2023 (CEST)

3:00 - 4:00pm Jize Yu (Chinese University of Hong Kong)

Title: Gaitsgory's central functor and Arkhipov-Bezrukavnikov's equivalence for p-adic groups

Abstract: In 2002, Arkhipov and Bezrukavnikov established an equivalence between the Iwahori-equivariant derived category of constructible \ell-adic sheaves and the Langlands dual group equivariant derived category of coherent sheaves on the Langlands dual Springer resolution for a connected reductive group over \bar{\mathbb{F}}_p. In this talk, we discuss this equivalence for p-adic groups by constructing a mixed-characteristic Gaitsgory's central functor. This is a joint work with J. Ansch\"utz, J. Louren\c{c}o, and Z. Wu.

May 25, 2023 (CEST)

3:00 - 4:00pm **Longke Tang **(Princeton University)

Title: The P^1-motivic cycle map

Abstract: ** **Recently, Annala, Hoyois, and Iwasa have defined and studied the P^1-motivic homotopy theory, a generalization of A^1-motivic homotopy that does not require A^1 to be contractible, but only requires pointed P^1 to be invertible. This makes it applicable to cohomology theories where the reduced cohomology of A^1 is nontrivial but that of P^1 is invertible, e.g. Hodge cohomology, de Rham cohomology, and prismatic cohomology. I will recall some basic facts in P^1-motivic homotopy theory, and construct the P^1-motivic cycle map, thus giving a uniform construction for the cycle maps of the above cohomology theories. If time permits, I will also use this cycle map to prove prismatic Poincaré duality.

May 18, 2023 (CEST)

3:00 - 4:00pm Kazuma Ohara (University of Tokyo)

Title: Types for Bernstein blocks and their Hecke algebras

Abstract: Let G be a connected reductive group defined over a non-archimedean local field F. The category R(G) of smooth complex representations of G(F) decomposes into the product of full subcategories Rˢ(G), called Bernstein blocks. In many cases, a block Rˢ(G) is equivalent to the category of modules over a C-algebra. More precisely, if there is a good pair (K, p) of a compact open subgroup K of G(F) and its irreducible smooth representation p, called an s-type, the block Rˢ(G) is equivalent to the category of modules over the Hecke algebra H(G, p) associated with the type (K, p).

In this talk, I will introduce several kinds of types; depth-zero types, tame

supercuspidal types, and Kim-Yu types. After that, I will explain the structure of the

Hecke algebras associated with these types. In particular, I will explain the following:

・The Hecke algebra associated with a depth-zero type is isomorphic to an extension of an

Iwahori-Hecke algebra by a twisted group algebra.

・For more general type, we can construct an isomorphism between its Hecke algebra and the

Hecke algebra associated with some depth-zero type.

This talk contains a joint work with Jeffrey Adler, Jessica Fintzen, and Manish Mishra.

May 4, 2023 (CEST)

3:00 - 4:00pm **Heejong Lee** (University of Toronto)

Title: Emerton-Gee stacks for GSp4 and Serre weight conjectures

Abstract: In the Langlands program, we want to construct a certain correspondence between automorphic representations and Galois representations. The meaning of this correspondence can be explained in terms of the L-functions. However, one can also ask how the structure of one side is reflected on the other side. Serre weight conjectures explicitly explain that how (Serre) weights of the automorphic side and the ramification behavior on the Galois side are related.

I will start my talk by giving some examples and heuristic arguments for the Serre weight conjectures. This will motivate us to understand certain Galois deformation rings. Then I will discuss Emerton-Gee stacks (which allows a more geometric approach to Galois representations) and local models of Le-Le Hung-Levin-Morra (which can describe parts of Emerton-Gee stacks explicitly), as well as their generalizations to the group GSp4.