The trace formula for GL2 and applications
This term (Easter 2022), Lukas Kofler and I are organizing a study group on the Selberg trace formula for GL2 and its applications to the Jacquet-Langlands correspondence.
[12/05 - MR11] Overview
[19/05 - MR11] Automorphic forms and representations
(Cuspidal) automorphic forms and representations on GL2
and quaternion algebras.
Statement of the (local and global) Jacquet-Langlands correspondences and definitions of the terms involved.
There are many references for this, for example §4.1 of Toby Gee's Notes
[26/05 - MR11] The trace formula for compact quotients
Brief overview of the relevant concepts from functional analysis.
Statement and proof of the trace formula in the case of compact quotients.
Problems in the non-compact case.
Lecture 12 of the Stanford Seminar
, §3 and 6 of David Whitehouse's notes
[02/06 - MR20] The trace formula for non-compact quotients: geometric side
Proof of the geometric side of the trace formula.
§7 of David Whitehouse's notes
§1 and 4.4 of Lecture 14 and 15 of the Stanford Seminar
[09/06 - MR20] Eisenstein series and the continuous spectrum
Eisenstein series and their analytic continuation.
Description of the continuous spectrum.
§8.1 and 8.2 of David Whitehouse's notes
, §2 and 3 of Lecture 14 and 15 of the Stanford Seminar
Corvallis §3, 4, 5
[16/06 - MR11] The trace formula for non-compact quotients: spectral side
Proof of the spectral side of the trace formula. Proof of a simple form of the trace formula.
§8.3 and 9 of David Whitehouse's notes
, the rest of §4 of Lecture 14 and 15 of the Stanford Seminar
and also 22 of the Stanford Seminar
Corvallis the rest of §6 and 7
[23/06 - MR11] Proof of Jacquet-Langlands
Recollection of the statement of local and global Jacquet-Langlands correspondences.
§10 of David Whitehouse's notes
, Lecture 20 of the Stanford Seminar