Wenhan Dai

Counting Points on Shimura Varieties

This short course is an introduction to the problem of computing the zeta function of Shimura varieties, or more concretely counting points. We will discuss the ideas of Langlands and Kottwitz of comparing the point counting with trace formulas, which eventually relates the zeta function with automorphic L functions. We will start with the basics and the historic roots, and in the end talk about some more recent developments for general abelian-type Shimura varieties. The course consists of six lectures, each 1.5 hours long. It is a natural follow-up of Liang Xiao’s short course in late July at BICMR.

• Lecturer: Yihang Zhu (University of Maryland, College Park).
• Time: August 9-20, 2021.
• Venue: Online (Zoom ID: 860 6562 2199; Code: 822154).

Notes for lectures

The latest version (edited by the lecturer) of the TeXed notes can be found here. See also the live-TeXed version by Liang Xiao.

The recoding videos collection is available at the BICMR Bilibili account. One-button-triplet (一键三连) is warmly appreciated!

• Lecture 1: Hasse-Weil zeta functions. (notes, video)
• Lecture 2: Integral models and the counting formula. (notes, video)
• Lecture 3: General way for computing (I). (notes, video)
• Lecture 4: General way for computing (II). (notes, video)
• Lecture 5: General counting formula and the trace formula. (notes, video)
• Lecture 6: Stabilization and proof of PCF in Hodge-type. (notes, video)