Spring 2020   Reading Seminar on Shimura Varieties

This is the webpage for the reading seminar on the geometry of Shimura varieties. The ultimate goal is to understand Helm's paper on geometric construction of Jacquet-Langlands correspondence in the case of U(2)-Shimura varieties. Along the way, we will touch base with topics including PEL type Shimura varieties and its mod p fibers, (partial) Hasse invariants, mod p (Hilbert) modular forms, Katz modular forms, overconvergent (Hilbert) modular forms, Grothendieck-Messing-Serre-Tate theorem on deformation of abelian varieties. Prerequisite of this seminar includes algebraic geometry, modular forms and modular curves, and basic sense of Langlands correspondence.

We meet every week on Thursday 3-5pm, starting on April 30, 2019.

The plan for this semester is as below; you can also find it in this file.

We will provide some exercises associated with each talk, following the links below. To keep up with the fast pace of this seminar, we strongly encourage the students to try to solve them, at least to look at them. Hints are always appended at the end of each file. Please pardon us for inaccuracy and errors in the notes and exercises. Comments are very welcome!

Click on "Show Abstract" to expand the abstract and click on "Hide Abstract" to hide it.




Topic and Abstract

April 30

Liang Xiao

Talk 1: Introduction to modular curves and Siegel modular varieties. (notes, exercise)
(Show Abstract)

May 7

Yibo Ji

Talk 2: Hilbert and Unitary modular varieties and geometric Hilbert modular forms. (speaker's notes, notes, exercise)
(Show Abstract)

May 14

Daming Zhou

Talk 3: Background material in algebraic geometry. (notes, exercise)
(Show Abstract)

May 21

Jiedong Jiang

Talk 4: Geometric properties of modular forms. (speaker's notes, notes, exercise)
(Show Abstract)

May 28

Longke Tang

Talk 5: Overconvergent modular forms. (notes, exercise)
(Show Abstract)

June 4

Liang Xiao

Talk 6: Arithmetic compactification of Hilbert modular varieties. (notes, exercise)
(Show Abstract)

June 11

Yupeng Wang

Talk 7: General theory of Shimura varieties. (speaker's notes, notes, exercise)
(Show Abstract)

June 18

Chunlin Shao

Talk 8: Dual BGG complex and Hodge theory. (notes, exercise)
(Show Abstract)

June 25

Weixiao Lu

Talk 9: Automorphic representations and cohomology of automorphic vector bundles. (speaker's notes, notes)
(Show Abstract)

July 2

Ruochuan Liu

Talk 10: Etale cohomology of Shimura varieties
(notes) (Show Abstract)

July 9

Yibo Ji

Talk 11: Shimura varieties of PEL type d'apres Kottwitz.
(speaker's notes, notes) (Show Abstract)

July 16

Jingbang Guo

Talk 12: Goren-Oort stratification.
(Show Abstract)

July 23

Ruiqi Bai

Talk 13: Description of sparse Goren-Oort strata.
(Show Abstract)

July 30

Zhixiang Wu

Talk 14: Geometry of unitary Shimura varieties with Iwahori level structure.
(Show Abstract)

August 6

Ruiqi Bai

Talk 15: Geometric construction of Jacquet-Langlands correspondence.
(Show Abstract)