Our aim in this seminar is to understand the main ideas of Theorem 1.5 in [Li-Liu]. It states that the non-vanishing of a certain L-function derivative implies the existence of certain non-trivial cycle classes. This should be viewed as a higher-dimensional generalization of the rank 1 case of the BSD Conjecture.
The proof in [Li-Liu] is via an arithmetic variant of the doubling method. A key input here is the theorem of Li-Zhang about intersection numbers of special divisors. For this reason, our main intermediate goal is to learn about [Li-Zhang].
Our seminar has been inspired by a similar seminar at MIT.
We meet in Room 203 on Tuesdays, 10:30 -- 12:00.
| Date | Topic | Contents | Speaker |
|---|---|---|---|
| March 4 | First meeting | Program introduction, seminar organization | |
| March 11 | Siegel-Weil formula | Introduce the Weil representation, theta lifting and the Siegel-Weil formula. [Li, §2.1-§2.3], [KR, §7-§8], and MIT seminar notes |
Chi Zhuoni |
| March 18 | Rallis inner product formula | Define the doubling L-function and explain the Rallis inner product formula. [Li, §2.4], [Li-Liu, §3], and MIT seminar notes |
Ying Hao |
| March 25 (?) | Unitary Shimura varieties | *Define unitary Shimura varieties. State their most basic properties. [Li, §3.1], [KR, §2-§4], and MIT seminar notes |
TBA |
| April 1 (?) | Special cycles | *Introduce special cycles. Show that Z(T) is entirely contained in the supersingular locus when T is n-by-n and non-degenerate. [KR, §2 and §5-§6] |
TBA |
| April 8 (?) | Arithmetic Siegel-Weil formula | *Define intersection numbers of local special divisors. Explain the main result of [Li-Zhang]. [Li-Zhang, §2.1-§2.4], and [KR, §11] |
Xiong Liangyi |
| April 15/22/29 (?) | Arithmetic doubling | Topics to be discussed: Beilinson's l-heights, projection to chomologically trivial cycles, application of arithmetic Siegel-Weil to computation of local indices, sketch of proof of [Li-Liu, Theorem 1.5]. References: [Li-Liu], and [Li] |
Mihatsch A., Yao Haodong |
* Feel free to work with an imaginary-quadratic extension over Q for simplicity
| [KR] | S. Kudla, M. Rapoport, Special cycles on unitary Shimura varieties II: Global theory, J. Reine Angew. Math. 697 (2014), 91–157. |
| [Li] | C. Li, Geometric and arithmetic theta correspondences, Lecture notes for the 2022 IHES Summer School |
| [Li-Liu] | C. Li, Y. Liu, Chow groups and L-derivatives of automorphic motives for unitary groups, Ann. of Math. 194 (2021), no. 3, 817-901. |
| [Li-Zhang] | C. Li, W. Zhang, Kudla-Rapoport cycles and derivatives of local densities, J. Amer. Math. Soc. 35 (2022), no. 3, 705–797. |