This seminar will be a mix of learning and research, on the topic of geometry of Shimura varieties mod p. We mainly follow some work by David Helm, Yichao Tian, and Liang Xiao. The purpose of this semester-long seminar is to go over basic facts and constructions around the concept of automorphic forms, especially its p-adic aspects: eigencurves, overconvergent automorphic forms. Most talks will be given by participants. For each talk, please prepare 90 minutes talk, followed by a 30 minutes exercise/discussion time. Talks will be given in English.
(2/28) Ruiqi Bai: Following [HTX] Introduce the moduli problem following [HTX, §2] and then discuss basic structures on a Shimura varieties [HTX, §3]. (video1, video2, notes)
(3/7) Deding Yang: Construction of cycles on Shimura varieties mod p, following [HTX, §4] up to Proposition 4.8. (video1, video2, notes)
(3/14) Ruiqi Bai: Go back to discuss expected cohomology of Shimura varieties, following [HTX, §2] and [Ko]. (video1, video2, notes)
(3/21) Zeyu Wang: Basics on intersection theory: summarize some main points from Fulton’s book [Fu, § 1-6] that is needed for the computation (e.g. sometimes, just state the main theorems and omit the proofs). (video1, video2, notes)
(3/28) Zeyu Wang: Computation of intersection number on DL varieties, following [HTX, §5].
(4/4) National Qing Ming Holiday.
(4/11) Hao Fu: Putting everything together, following [HTX, §6]. No spending too much time on [HTX, Proposition 6.4]. (video1, video2, video3, notes)
(4/18) Wenhan Dai: Goren–Oort strata for U(2)-Shimura varieties: sparce case, following [He], up to section 4. (video1, video2, notes)
(4/25) Liang Xiao & Wenhan Dai: First Wenhan Dai finishes the previous talk. Then Liang talks about Carayol’s construction. (video1, video2, notes)
(5/2) National Labor-Day Holiday.
(5/9) Haocheng Fan: Goren–Oort strata for U(2)-Shimura varieties: general case I, following [TX, §4 and §5]. Translate the discussion in [TX] into Helm’s setup.
(5/16) Haocheng Fan: Goren–Oort strata for U(2)-Shimura varieties: general case II, following [TX, §4 and §5]. Translate the discussion in [TX] into Helm’s setup.
(5/23) Deding Yang: Ampleness of automorphic line bundles on U(2)-Shimura varieties [TX, §6].
(5/30) Ruiqi Bai: Ihara’s lemma vs. geometric level-raising.
(6/6) Hao Fu: Level-raising for U(3)-Shimura varieties I.
(6/13) Hao Fu: Level-raising for U(3)-Shimura varieties II.
(TBD) TBD: Galois representations and torsion in the coherent cohomology of Hilbert modular varieties, follow [ERX].
The videos for this seminar series are valid until 2022-09-30 23:59.
References
[ERX] Emerton, Reduzzi, Xiao, Galois representations and torsion in the coherent coho- mology of Hilbert modular varieties, Crelle’s journal.
[Fu] W. Fulton, Intersection theory, book.
[He] D. Helm, A geometric Jacquet–Langlands correspondence for U(2) Shimura varieties, Israel J. Math. 187 (2012), 37–80.
[HTX] D. Helm, Y. Tian, and L. Xiao, Tate cycles on some unitary Shimura varieties mod p, Algebraic Number Theory 11 (2017), 2213–2288.
[Ko] R. Kottwitz, On the λ-adic representations associated to some simple Shimura varieties, Invent. Math. 108 (1992), 653–665.
[TX] Y. Tian and L. Xiao, On Goren-Oort stratification for quaternionic Shimura varieties, Compositio Mathematica 152 (2016), 2134–2220.