## Mod p Representations of GL_2 and of Quaternion Algebras

**Speaker(s): ** Yongquan Hu(AMSS, MCM)

**Time: **November 23 - December 21, 2021

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

**Time: 13:00-14:30 & 15:00-16:30 23/11, 30/11, 7/12, 14/12, 21/12**

This short course aims to give an introduction to some recent progress on mod p Langlands program and mod p Jacquet-Langlands program. We will mainly focus on the case of GL_2(Q_p).

We will discuss the following topics:

(i) Mod p representation theory of GL_2(Qp) and of quaternion algebra over Qp;

(ii) Serre weights in the quaternion algebra setting;

(iii) Galois deformation theory;

(iv) Gelfand-Kirillov dimension of admissible representations of p-adic groups;

(v) Scholze’s functor and some vanishing result.

Prerequisite:

Basic representation theory of p-adic groups; basic facts on local fields and p-adic analytic groups.

References:

1. C. Breuil, Representations of Galois and of GL_2 in characteristic p, course at Columbia University.

2. C. Cheng, Mod p representations of local division algebras over p-adic fields, unpublished notes.

3. C. Khare, A local analysis of congruences in the (p,p) case. II, Invent. Math. 143 (2001), 129-155.

4. P. Scholze, On the p-adic cohomology of the Lubin-Tate tower, Ann. Sci. ENS, 51 (2018), 811-863.

5. V. Paskunas, On the Breuil-Mezard conjecture, Duke Math. J. 164 (2015), 297-359.

6. V. Paskunas, On some consequences of a theorem of Ludwig, arXiv: 1804.07567