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.
|
Dates |
Speaker |
Topic and Abstract |
|---|---|---|
|
April 30 |
Liang Xiao |
Talk 1: Introduction to modular curves and Siegel modular varieties. (notes, exercise) |
|
May 7 |
Yibo Ji |
Talk 2: Hilbert and Unitary modular varieties and geometric Hilbert modular forms. (speaker's notes, notes, exercise) |
|
May 14 |
Daming Zhou |
Talk 3: Background material in algebraic geometry. (notes, exercise) |
|
May 21 |
Jiedong Jiang |
Talk 4: Geometric properties of modular forms. (speaker's notes, notes, exercise) |
|
May 28 |
Longke Tang |
Talk 5: Overconvergent modular forms. (notes, exercise) |
|
June 4 |
Liang Xiao |
Talk 6: Arithmetic compactification of Hilbert modular varieties. (notes, exercise) |
|
June 11 |
Yupeng Wang |
Talk 7: General theory of Shimura varieties. (speaker's notes, notes, exercise) |
|
June 18 |
Chunlin Shao |
Talk 8: Dual BGG complex and Hodge theory. (notes, exercise) |
|
June 25 |
Weixiao Lu |
Talk 9: Automorphic representations and cohomology of automorphic vector bundles. (speaker's notes, notes) |
|
July 2 |
Ruochuan Liu |
Talk 10: Etale cohomology of Shimura varieties |
|
July 9 |
Yibo Ji |
Talk 11: Shimura varieties of PEL type d'apres Kottwitz. |
|
July 16 |
Jingbang Guo |
Talk 12: Goren-Oort stratification. |
|
July 23 |
Ruiqi Bai |
Talk 13: Description of sparse Goren-Oort strata. |
|
July 30 |
Zhixiang Wu |
Talk 14: Geometry of unitary Shimura varieties with Iwahori level structure. |
|
August 6 |
Ruiqi Bai |
Talk 15: Geometric construction of Jacquet-Langlands correspondence. |