P.S. Class of 10/10 will change to 10/11(Wed) still 10:15-11:45 and same classroom Room 609.

Contents:

Abelian varieties and Shimura varieties provide rich sources for current research in Arithmetic Geometry. This course will discuss some topics on abelian varieties, and Shimura varieties as well as related topics on algebraic number theory, algebraic groups and representation theory. It is roughly divided into Tuesday session (require more background, in English) and Friday session (more elementary and explicit, in Chinese).

The Tuesday session will focus on the following: supersingular and supespecial abelian surfaces over finite fields as well as their classifications, methods for counting abelian varieties; Drinfeld modules, supersingular Drinfeld modules, algebraic Drinfeld modular forms, Hasse invariants, v-rank strata and connectedness; unimodular Hermitian forms, Siegel’s and exact mass formulas, computation of local densities; integral models of Shimura varieties and geometry of its special fibers: definition of stratifications and constructions, geometric meaning and group-theoretic meaning; CM algebra and endomorphism algebras of abelian varieties over finite fields. We also hope to discuss the cohomology of the supersingular or superspecial locus and algebraic modular forms. 

The Friday session will focus on the following: arithmetic of definite quaternion algebras over totally real fields, reduced unit groups of maximal orders and their classification and distribution; arithmetic invariants for totally imaginary quadratic extension of totally real fields, class number parity; introductory integral representations of finite groups or orders in semi-simple algebras, structures on algebraic tori.

References:

1. Abelian varieties: Mumford’s book

2. Shimura varieties: Deligne: 2 papers, Milne: Introduction, Kottwitz: Isocrystals in Compos. Math (1985, 1997), JAMS (1992), Kisin JAMS(2013,2017), Rapoport-Zink’s book, He-Rapoport, Gortz-He, Gortz-He-Nie, Pappas-Zhu, Pappas-Kisin, Riemann’s book: Quaternionic Shimura varieties. Helm’s paper on cohomology of supersingular locus, Varsharsky’s paper on moduli spaces of F-sheaves. 

3. Mass formulas: Hashimoto and Koseki, Tohoku 41 (1989), 1-30. Gan-JK Yu-Hanke Duke J. Math (2002).

4. Drinfeld modules: Drinfeld’s 2 papers, Laumon’s 2 books, Pink Compacfication of Drinfeld moduli schemes Manu Math (2013), Pink and Schieder, JAG (2013).

5. Curtis and Reiner’s books; Conner and Hurrelbrink: Class number parity; Springer: Linear Algebraic Groups.

Contact: Chia-Fu Yu for more detailed references.