Abstract:

Lecture 1: Endoscopic classification for classical groups
We give an introduction to Arthur and Mok’s work on endoscopic classification of representations of classical groups. Locally, this gives the local Langlands correspondence for quasi-split classical groups over non-Archimedean local fields. Globally, this gives a description of the discrete spectrum for quasi-split classical groups over number fields. We recall R. Chen and J. Zou’s work extending these results to non-quasi-split cases, and prove the endoscopic character identity. These identities will be crucial in the second lecture.

Lecture 2: Fargues—Scholze parameters and compatibility.
Fargues—Scholze has constructed a new candidate for the semisimple (infinitesimal) version of the local Langlands correspondence, for general reductive groups over non-Archimedean local fields. Using the endoscopic character identity, we can partially understand the supercuspidal part of the cohomology of local Shtuka spaces. Using basic uniformization results for Shimura varieties, we relate cohomology of local and global Shimura varieties. Finally, using Langlands—Kottwitz methods, we prove that Fargues—Scholze’s construction is compatible with the classical one in lecture 1 for unramified orthogonal and unitary groups. This result can be used to prove torsion vanishing results for orthogonal and unitary Shimura varieties, though it has been recently proved by X. Yang and X. Zhu for general Abelian type Shimura varieties using Zhu’s categorical local Langlands. This torsion vanishing result is crucial for the application in the third lecture.

Lecture 3: Theta correspondence and Beilinson—Bloch—Kato conjecture.
The Beilinson—Bloch—Kato conjecture is a far-fetching generalization of the (rank part of the) BSD conjecture for modular elliptic curves. For elliptic curves over Q, the rank part of the BSD conjecture has been proved by Kolyvagin and Gross—Zagier when analytic rank is at most one. Their proof relies on certain period formula. higher generalizations, including the Gan—Gross—Prasad conjecture and Rallis inner product formula, are available for motives related to unitary and orthogonal groups. Using Galois-theoretic argument of Kolyvagin and Y. Liu, BBK conjecture is partially proved for U(N)*U(N+1)-motives by Y. Liu, Y. Tian, L. Xiao, W. Zhang, and X. Zhu. Using theta correspondence, we prove that their result implies the BBK conjecture for U(2n)-motives in the analytic rank 0 case. Similar trick works in the orthogonal case. If time permits, we talk about the work in progress proving the BBK conjecture for O(N)*O(N+1)-motives when analytic rank is at most one.

Time:

Aug 5/7/9, 2:00-3:30 PM