p-adic Day II
Time: 2025-10-28
Published By: He Liu
Time: November 4, 2025
Venue: Room 77201, Jingchunyuan 78, BICMR
9:00-10:00 Yang Pei (Peking University)
Title: Coverings of the Drinfeld upper plane and categorical p-adic Langlands
Abstract: We generalize to all levels of the tower of coverings of the Drinfeld upper plane an exact sequence established by Lue Pan for the first covering. Furthermore, we introduce two functors, inspired by the categorification of the p-adic local Langlands correspondence, in Banach and locally analytic versions respectively. We then compute the sheaves associated with the representations appearing in our sequence. As an application, we show that all proper quotients of the universal unitary completion of a supercuspidal representation have finite length.
10:30-11:30 Xiaozheng Han (Peking University)
Title: p-adic Hodge parameters in non-critical crystalline representations valued in GSp4
Abstract: For a given crystalline representation V of the absolute Galois group of Qp, which is valued in GSp4 and satisfies certain generality conditions, I constructed a locally analytic representation π_1(V) of GSp4(Q_p), encoding all the information of V. By an argument of the local-global compatibility under certain settings, I proved that π_1(V) is a subrepresentation of the locally analytic representation corresponding to V in the sense of p-adic LLC. This generalizes the result on GLn of Yiwen Ding in the last year.
11:30-13:00 Lunch&Discussion
13:00-14:00 Stefano Morra (Université de Paris Saint-Denis)
Title: Finite length for unramified GL2.
Abstract: Let $p$ be a prime number and $K$ a finite unramified extension of$\mathbb{Q}_p$. The smooth $\mathrm{GL}_2(K)$ representations appearing in the mod $p$ local Langlands program are expected to satisfy desirable properties, in particular their "structure" should be predicted by the corresponding 2-dimensional mod $p$ representations of $\mathrm{Gal}(\overline{K}/K)$ (e.g. they are irreducible if and only if the local Galois representation is).
In joint work with C. Breuil, Y. Hu, F. Herzig, B. Schraen we show that the smooth mod p representations $\pi$ of $\mathrm{GL}_2(K)$ appearing in Hecke eigenspaces of the cohomology of Shimura curves are of finite length and satisfy several further constraints coming from the structure of the local Galois representation.
In this talk we focus on the special case when $K$ is a quadratic extension of $\mathbb{Q}_p$, and introduce the main tools which appear in the proof of the general case, such as (\varphi,Gamma)-modules and the Iwahori socle filtration of $\pi$.
This is joint work with C. Breuil, F. Herzig, Y. Hu and B. Schraen.
14:30-15:30 Zhongyipan Lin (Tongji University)
Title: Potentially diagonalizable lifts of Langlands parameters in types ABCDG
Abstract: In this talk, I will discuss several motivations for asking whether de Rham lifts of mod p local Langlands parameters exist unconditionally, explain the cases I have established over the years:
* ramified and unramified unitary groups (type A), p > 2
* GSpin (type B & D) and GSp (type C), p > 2
* Chevalley G2 (type G), p > 3, and touch on some applications.
