## P-adic day

**Time: **April 19, 2018

**Venue: ** Room 77201, Jingchunyuan 78, BICMR

**Organizer: **

Yiwen Ding (Peking University)

Ruochuan Liu (Peking University)

**Speaker: **

Laurent Berger (UMPA, ENS de Lyon)

Yongquan Hu (Morningside center, AMSS)

Sandra Rozensztajn (UMPA, ENS de Lyon)

Benjamin Schraen (CNRS, Ecole polytechnique)

Xu Shen (Morningside center, AMSS)

**Schedule: **

09:30-10:30 Yongquan Hu

10:45-11:45 Benjamin Schraen

*11:45-13:30 Lunch break*

13:30-14:30 Sandra Rozensztajn

14:45-15:45 Xu Shen

16:00-17:00 Laurent Berger

**Title & ****Abstract:**

Laurent Berger,

Title: P-adic Langlands and Lubin-Tate (phi,Gamma)-modules

Abstract: In order to generalize the p-adic Langlands correspondence to GL_2(F), where F is a finite extension of Q_p, it seems useful to have theory of Lubin-Tate (phi,Gamma)-modules. I will discuss two ways of doing this and compare them.

Yongquan Hu,

Title: Asymptotic growth of the cohomology of Bianchi groups

Abstract: Given a level N and a weight k, we know the dimension of the space of (classical) modular forms. This turns out to be unknown if we consider Bianchi modular forms, that is, modular forms over imaginary quadratic fields. In this talk, we consider the asymptotic behavior of the dimension when the level is fixed and the weight grows. I will first explain an upper bound obtained by Simon Marshall using Emerton’s completed cohomology and the theory of Iwasawa algebras. Then I explain how to improve this bound using the mod p representation theory of GL2(Qp).

Sandra Rozensztajn,

Title: On the reduction modulo p of crystalline representations of dimension 2

Abstract: I will talk about the problem of studying the reduction modulo p of crystalline representations of dimension 2 of the Galois group of Q_p. In particular, I will be interested in the following situation: fix Hodge-Tate weights and a residual representation, and consider the locus parametrizing crystalline representations with the given weights and reduction modulo p. What can be said about this locus in general?

Benjamin Schraen,

Title: Density of automorphic points in polarized global deformation spaces.

Abstract: In the 90’s Gouvea and Mazur proved that the Galois representations that are (up to twist) associated to modular forms are Zariski-dense in the generic fiber of certain Galois deformation rings. This result was generalized to 3-dimensional polarized Galois representations by Chenevier, using the same strategy involving the so-called 〝infinite fern〞. I will report on joint work with Eugen Hellmann and Christophe Margerin concerning generalizations of this statement to arbitrary dimensions. This builds upon the analysis of the geometry of a space of trianguline local p-adic Galois deformations and the construction of companion points on eigenvarieties.

Xu Shen,

Title: P-adic period domains and the Fargues-Rapoport conjecture

Abstract: Given a local Shimura datum, thanks to the recent progress in p-adic Hodge theory, now one knows how to construct the attached p-adic period domain and the tower of local Shimura varieties over it. In this talk, we explain some ideas in the recent joint work with Miaofen Chen and Laurent Fargues on the structure of some p-adic period domains. More precisely, we will sketch a proof of the Fargues-Rapoport conjecture: for a basic local Shimura datum (G,b,µ), the weakly admissible locus coincides with the admissible one if and only if the Kottwitz set B(G, µ) is fully Hodge-Newton decomposable.