Schedule for: 25w5400 - New Tools in the Local and Global Langlands Programs

A set dinner is served daily between 5:30pm and 7:30pm in the Xianghu Lake National Tourist Resort.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

In my joint work with Scholze we defined a spectral action of perfect complexes on the moduli of local Langlands parameters on the derived category of some l-adic sheaves on Bun_G, the moduli of G-bundles on the curve. Applied to a so-called Whittaker sheaf this spectral action defines a functor, and we expect this functor extends to a categorical equivalence. Nevertheless, since the Whittaker representation is only defined over a cyclotomic extension this functor is not defined rationally. I will explain how to define a Gm-gerbe over the moduli of local Langlands parameters that allows us to make this functor rational, by replacing perfect complexes by the one twisted by this Gm-gerbe (in the sense of Lieblich). The construction is done in the more general setting of the motivic spectral action, and in the meantime we obtain some interesting results about this motivic spectral action. This moreover works in the setting of the Emerton-Gee stack. This is joint work in progress with Naoki Imai.

For G a connected reductive group over the p-adic numbers, the categorical local Langlands correspondence seeks to relate the category of \ell-adic sheaves on the moduli stack of G-bundles on the Fargues-Fontaine curve with the category of coherent sheaves on the moduli stack of Langlands parameters. The former category is built from considering smooth representations of varying inner forms J_{b} of G indexed (roughly) by Harder-Narasimhan polygons b together with gluing data indexed by the natural partial ordering on such polygons. While embedding the category of representations of J_{b} for fixed b as coherent sheaves on the stack of L-parameters can be understood through classical local Langlands and enhancements such as the Bezrukavnikov equivalence, explicating the gluing datum remains much more mysterious. The goal of this talk is to do precisely this; in particular, we will explain how in the specific case where G = SL_{2}, one can describe how the Whittaker models of the different representation categories can glue together, verifying a form of constant term compatibility under the categorical equivalence in the process. This is joint work with Naoki Imai.

Lunch is served daily between 11:30am and 1:30pm in the Xianghu Lake National Tourist Resort

Classical Fourier theory describes measures on a locally compact abelian group in terms of functions on its Pontryagin dual. In this talk, I will explain an analogous theory for (families of) p-divisible rigid analytic groups and their duals that recovers the Amice transform when applied to the open unit disk considered as multiplicative group. If time permits, I will explain how this is related to the study of families of p-adic automorphic forms. This is joint work with Andrew Graham and Sean Howe.

Igusa stacks are p-adic geometric objects that roughly parametrize abelian varieties up to isogeny. In a joint work with Daniels, van Hoften, and Zhang, we constructed Igusa stacks for Hodge type Shimura data, and discussed how its cohomology relates to the cohomology of Shimura varieties. But there is another pleasant feature: Igusa stacks are unique if they exist, and automatically functorial with respect to morphisms of Shimura data. I will discuss this result as well as some geometric input that goes into the proof.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

I will outline how to establish the so-called geometric principal series part of the categorical local Langlands correspondence, which will extend my previous work on the tame part. Joint work in progress with Charlotte Chan and Tasho Kaletha.

We prove a torsion vanishing result for the generic part of the cohomology of Shimura varieties of abelian type, generalizing the results of Caraiani-Scholze, Koshikawa, Hamann-Lee, and others. Our proofs utilize the unipotent categorical local Langlands correspondence developed by Zhu. We also discuss a generalization of the vanishing results to the Iwahori-Whittaker coefficients of the cohomology of Shimura varieties. This is a joint work with Xinwen Zhu.

I will report on my upcoming collaboration with Linus Hamann, Alexander B. Ivanov, João Lourenço and Konrad Zou. In this project we compare two constructions of the automorphic side of the categorical Langlands correspondence. On one side of the comparison is the category D_et(BunG) considered by Fargues--Scholze, on the other side is the category Shv(B(G)) considered by Zhu. We 1) outline the construction of the functor and 2) explain the overall strategy to show that it is an equivalence. Both 1) and 2) depend on geometric results due to G., I. and Zillinger.

In this talk, I would like to discuss a comparison between two kinds of representations of p-adic reductive groups arising from different origins. One is Yu's algebraic construction, further developed by Kim-Yu and Kaletha. The other is a geometric construction recently established by Chan-Ivanov, which generalizes the classical Deligne-Lusztig construction to the positive-depth setting. Our basic strategy is to compare the trace characters of those two representations. A central idea in our argument is to introduce an analogue of the classical Green function (in the context of Deligne-Lusztig theory) for both the algebraic and geometric representations, which enables us to get a character formula completely parallel to the classical Deligne-Lusztig character formula. This is joint work with Charlotte Chan (University of Michigan).

TBA

In this talk, we discuss the Langlands correspondence for reductive groups over finite fields, which we call the finite Langlands correspondence. We discuss also its conjectural relation with the categorical local Langlands correspondence. This is partially based on a joint work with David Vogan.

I will report on my joint work in progress with Lue Pan which shows that, in the rational p-adic completed cohomology of a general Shimura variety, "sufficiently regular" infinitesimal weights (whose meaning will be explained) can only show up in the middle degree. I will also mention some byproducts and explain the main ingredients in our work, if time permits.

I will explain how to construct examples of exotic Hecke correspondences between the special fibers of different Shimura varieties, at possibly ramified primes. These can be used to construct geometric realizations of the Jacquet-Langlands correspondence, and verify generic instances of the Tate conjecture for these Shimura varieties, generalizing the work of Xiao-Zhu in the unramified case. The key will be to consider splitting models of Shimura varieties. This is work in progress.

I will report on joint work in progress with Pappas in which we outline a procedure to obtain local models for Shimura varieties with parahoric level, from those with Weil restriction of hyperspecial level. Key to this is a conjecture on the reducedness of the fixed point scheme of affine Schubert varieties under the action of a tame group. We give some evidence for this conjecture in the case of groups of type A, using the moduli description of such local models due to Görtz.

Affine Deligne-Lusztig varieties are a class of subschemes of affine flag varieties that play an important role in the description of the reduction modulo p of Shimura varieties. However, a complete description of their geometry is only available in very few cases, under the so-called fully Hodge-Newton decomposablility condition. I will explain how to relax this condition, still obtaining a good description of the geometry of the associated ADLV. (Joint work with Felix Schremmer)

We (re)construct the ordinary part of the eigenvariety for a reductive group G from the module of integral ordinary distributions. For G=U(n)xU(n+1), we construct the Bessel periods as H-invariant distributions, where H is the diagonal subgroup of G appearing in the Gan-Gross-Prasad conjecture. Finally we relate such period to the characteristic divisor of the Selmer group of the standard Galois representation over the eigenvariety, which we regard as the Iwasawa main conjecture over ordinary eigenvariety.

This talk will be on work in progress with Teruhisa Koshikawa. The aim of our project is to construct and study certain kinds of sheaves with representation-theoretic import on the stack of L-parameters for a local field. In particular, we define a category of cuspidal sheaves that will correspond under the categorical Langlands conjecture to a category of sheaves on BunG generated by supercuspidal representations. We also describe a construction of eigensheaves on the Galois side corresponding to Arthur parameters and describe some of their properties.