Note：The lectures will resume online on Nov.26, Dec.3, Dec.10,2022. 

Abstract: 

This series of talk focuses on constructing cycles on the special fibers of Shimura varieties, partially motivated by the Tate conjecture and the Langlands correspondence.  Roughly speaking, the cohomology of Shimura varieties carries information from both automorphic and Galois sides. I will first present the joint work with Xinwen Zhu showing that the Tate classes on the cohomology of the special fiber of Shimura varieties can be generated from the irreducible components of the basic locus, at least ``generically".   In later talks, I will mention some arithmetic applications of such construction, as well as some potential variants of this result.

The tentative plan for the series of talks is as follows.

Talk 1: Motivation from Tate conjecture and statements of the main results.
Talk 2: Geometric Satake equivalence and irreducible components of affine Deligne--Lusztig varieties.
Talk 3: Categorical trace construction and correspondences between Shimura varieties.
Talk 4: A short introduction to Xinwen Zhu's work on coherent sheaves on stack for Langlands parameters.
Talk 5: An introduction to Euler system argument and application to Beilinson--Bloch--Kato's conjecture.

Bio:

Liang Xiao is a Professor at Beijing International Center for Mathematical Research at Peking University, China. His research interest includes number theory, arithmetic geometry, and representation theory.

He received a bachelor degree in mathematics at Peking University, China, and obtained Ph.D in pure mathematics at MIT under the supervision of Kiran S. Kedlaya. After that, he was a Dickson Instructor at University of Chicago, and he later held faculty positions at University of California, Irvine, and at University of Connecticut.

Notes:

2022-10 Mini-course 1 - Motivation from Tate conjecture .pdf

2022-10 Mini-course 2 - Geometric Satake and irreducible components of ADLV.pdf

2022-10 Mini-course 3 - Categorical trace construction.pdf

2022-10 Mini-course 4 - Irreducible components of affine Deligne-Lusztig varieties.pdf

2022-10 Mini-course 5 - Coherent sheaves on moduli stack of Langlands parameters.pdf

2022-10 Mini-course 6 - Euler system and application to Beilinson-Bloch-Kato conjecture.pdf

Videos:

http://archive.ymsc.tsinghua.edu.cn/pacm_course?html=Cycles_on_special_fiber_of_Shimura_varieties_and_arithmetic_applications.html