## Summer School on Number Theory

**Speaker(s): ** Lue Pan, Zicheng Qian, Jun Su, Jingwei Xiao, Liang Xiao

**Time: **August 13 - August 15, 2018

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

Organizer:

Yiwen Ding (Peking University)

Ruochuan Liu (Peking University)

Speaker:

Lue Pan (Princeton University)

Zicheng Qian (Universite Paris-Sud)

Jun Su (Princeton University)

Jingwei Xiao (MIT)

Liang Xiao (University of Connecticut)

Schedule:

8.13

10:00-11:30 Jun Su

11:30-13:00 Lunch Break

13 :00-14 :30 Jingwei Xiao

14:30-15:00 Coffee Break

15 :00-16 :30 Zicheng Qian

8.14

9:00-10:30 Lue Pan

10:30-11:00 Coffee Break

11 :00-12 :30 Liang Xiao

12:30-13:30 Lunch Break

14:00-15 :30 Jun Su

15:30-16:00 Coffee Break

16 :00-17 :30 Jingwei Xiao

8.15

10 :00-11 :30 Zicheng Qian

13 :00-14 :30 Lue Pan

14:30-15:00 Coffee Break

15 :00-16 :30 Liang Xiao

Title & Abstract:

Jun Su

Title: Coherent cohomology of Shimura varieties and automorphic forms

Abstract: The canonical extensions of automorphic vector bundles on Shimura varieties are vector bundles whose global sections characterize holomorphic automorphic forms, and in this series of talks we show that the Hecke module structures on their higher cohomology also come from automotphic forms. As people have associate Galois representations to some eigenclasses in these cohomology, we obtain a bridge between some new classes of automorphic representations and Galois representations. In the first talk we’ll explain the motivation and consequences of the result. Our work is vastly inspired by an analogous characterization of the cohomology of automorphic local systems on locally symmetric spaces proved by Franke, and we’ll also talk about the relation and difference between the 2 situations. In the second talk we’ll sketch the proof and explain some main techniques.

Jingwei Xiao

Title: Jacquet-Rallis transfer and endoscopic fundamental lemma for unitary groups

Abstract: For each reductive group G, Langlands introduced a series of reductive groups H, called endoscopic groups of G. They are used to relate twisted orbit integrals on G to stable orbit integrals on H. Endoscopic groups appears when stabilizing the trace formula of G and also when counting points of special fibers of Shimura varieties. In the first part of this talk, we explain the hamonic analysis aspect of endoscopic groups. In the second part, we explain our new proof of the endoscopic fundamental lemma in the case of unitary groups. While previous proof requires passing to positive characteristic and using geometry, our proof is purely local hamonic analysis, and uses a surprising germ expansion identity in the Jacquet-Rallis transfer in characteristic 0.

Zicheng Qian

Title: mod p local global compatibility for GLn(Qp) in Fontaine-Laffaille case

Abstract: Let p be a prime number, n>2 an integer, and F a CM field in which p splits completely. Assume that a continuous automorphic Galois representation r : Gal(Q-/F) →GLn(Fp) satisfies certain genericity conditions at a place w above p, and that every subquotient of r|Gal(Qp-/Fw) of dimension>2 is Fontaine–Laffaille and sufficiently generic. We show that the isomorphism class of r|Gal(Qp-/Fw) is determined by GLn(Fw)-action on a space of mod p algebraic automorphic forms cut out by the maximal ideal of a Hecke algebra associated to r. In particular, we show that the wildly ramified part of r|Gal(Qp-/Fw) is determined by the action of Jacobi sum operators (seen as elements of Fp[GLn(Fp)]) on this space. This is a joint work with Chol park if r is upper triangular at w and joint work in progress with Bao V. Le Hung, Daniel Le, Stefano Morra and Chol Park in the general case. In the first talk, we will recall some background and the formulation of local global compatibility. In the second talk, we will give more details of the idea of proof of the theorem above.

Lue Pan

Title: An introduction to automorphy lifting theorems

Abstract: Automorphy lifting theorems (ALT) roughly say that under certain conditions, an l-adic Galois representation arises from an automorphic representation as long as it is congruent to some automorphic Galois representation. We will explain the classical ALT in the simplest example and an ALT in the setting of completed cohomology.

Liang Xiao

Title: Higher Hida theory of Pilloni

Abstract: I will report on a recent work of Pilloni which generalizes the classical Hida theory to the case of Siegel modular forms of non-regular weights. This is a crucial ingredient of the forthcoming work of Boxer-Calegari-Gee-Pilloni on potential modularity of abelian surfaces.