1. Generalized Kloosterman sheaves for reductive groups. by Daxin Xu (Morningside Center of Mathematics, Beijing)

Summary: Deligne constructed a remarkable local system on G_m attached to a family of Kloosterman sums. I will present the work of Heinloth-Ngô-Yun on the construction of generalized Kloosterman sheaves for reductive groups in the framework of geometric Langlands program. This gives an example of the geometric Langlands correspondence with wild ramification. I will also sketch the parallel constructions in the de Rham and p-adic settings based on the work of Frenkel-Gross, X. Zhu and my joint work with X. Zhu.

2. Introduction to p-adic L-functions. by Guhanvenkat Harikumar (Morningside Center of Mathematics, Beijing)

Summary: The aim of this special topics course will be to introduce the theory of p-adic L-functions which are the heart of Iwasawa Theory. We will focus primarily on the construction of the Kubota-Leopoldt p-adic L-function in this lecture series. We will also realize the Kubota-Leopoldt p-adic L-function as the constant term of a p-adic family of Eisenstein series. Time permitting, we will dwelve into the Iwasawa main conjecture relating the Kubota-Leopoldt p-adic L-function to the arithmetic of cyclotomic fields. The primary reference for this course, [RW], will be the lecture notes of Chris Williams (Warwick) and Joaquin Rodrigues (Lyon). Some other useful references are Lang's Cyclotomic Fields I & II [Lan90] and Pierre Colmez's M2 Lecture notes on the p-adic zeta function [Col].

SCHEDULE

REFERENCES

[Col] Pierre Colmez. La Fonction Zêta p-adique. Available online at https://webusers.imj-prg.fr/ pierre.colmez/Kubota-Leopodt.pdf.

[Lan90] Serge Lang. Cyclotomic fields I and II, volume 121 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1990. With an appendix by Karl Rubin.

[RW] Joaquin Rodrigues and Chris Williams. An introduction to p-adic l-functions. Available online at https://chriswilliams1404.wixsite.com/website/course-on-p-adic-l-functions.