The seminar will meet Wednesday 3:30-5:00PM+ in Van Vleck B211 .
The goal of this two-semester long semiar is to have a understand the sprits the relative trace formula and their applications. In the first semester, we will review the automorphic representations, study the regular trace formula, and prove the Jacquet-Langlands correspondence. In the second semester, we will go into Jacquet's proof of Waldspurger formula, then some further topics.
To receive the messages, please subscribe our google group by sending any message to 2024-2025_rtf-reading-seminar+subscribe@g-groups.wisc.edu. To get credits, please enroll Math941 SEM004, and give a talk in the seminar. You are encouraged to give more than one talks since we don't have enough speakers.
Sept. 11 | Zhiyu Zhang | Introduction to the relative trace formula. |
Sept. 18 | Ryan Tamura | Automorphic representations for \(\mathrm{GL}_2\). |
Sept. 25 | Phillip Harris | Representation of \(\mathrm{GL}_2(\mathbb{R})\). |
Oct. 2 | Chenghuang Chen | Representations of \(\mathrm{GL}_2(\mathbb{Q}_p)\). |
Oct. 9 | Alejo Salvatore | L-groups, L-functions, functoriality. |
Oct. 16 | Phillip Harris | Trace formula for compact quotient. |
Oct. 23 | Simon Marshall | Trace formula for non-compact quotient I. |
Oct. 30 | (Maybe still) Simon Marshall | Trace formula for non-compact quotient II. |
Nov. 6 | TBD | Trace formula for adelic quotient. |
Nov. 13 | TBD | Jacquet-Langlands correspondence I. |
Nov. 20 | TBD | Jacquet-Langlands correspondence II. |
Nov. 27 | Thanksgiving recess | |
Dec. 4 | TBD | Tanya Khovanova's recent work |
TBD | TBD | Hecke periods et al. |
TBD | TBD | Waldspurger formula and applications. |
TBD | TBD | Jacquet's proof of Waldspurger formula I. |
TBD | TBD | Jacquet's proof of Waldspurger formula II. |
TBD | TBD | Jacquet's proof of Waldspurger formula III. |
TBD | TBD | Jacquet's proof of Waldspurger formula IV. |
TBD | TBD | Jacquet's proof of Waldspurger formula V. |
TBD | TBD | Further topics I. |
TBD | TBD | Further topics II. |
TBD | TBD | Further topics III. |
TBD | TBD | Further topics IV. |
Abstract: In this talk, we will talk about relative trace formulas, which are equalities between period integrals (the spectral side) and relative orbital integrals (the geometric side) as distributions on compactly supported adelic functions for the reductive group \(G\). We will explain the general principle, and then focus on examples related to Gan-Gross-Prasad conjecture and Rankin-Selberg periods for \(G=GL_n \times GL_m\).
Defn of automorphic forms and automorphic representations: [B] Section 3.3, [GH2] ch 4,5
Tensor product theorem:[B]Section 3.4, [GH2]Ch 10
Strong approximations(and classical forms):[B]3.3
Cuspidality, Spectral decomposition: [B]3.3, [G]Ch5, [GH1]Ch9
Definition of admissible \((\mathfrak{g},K)\)-module
Classification of irreducible admissible \((\mathfrak{g},K)\)-module, principal series, and discrete series
Relation to automorphic forms (modular forms, Maass forms)
Definition of smooth, admissible representations
Parabolic induction, principal series, Supercuspidal, and Jacquet module
If time permitted, Satake isomorphism [GH1] 7.1, 7.2
Satake isomorphism, Satake parameter [GH1] 7.1. 7.2
Root data of reductive groups, L-groups, and Langlands L-functions
Weil(-Deligne) groups, local Langlands, L-packet, and functoriality [C] Lec 11
Base change problems in \(\mathrm{GL}_1\) and \(\mathrm{GL}_2\) [F] Introduction
Present it using Selberg's original paper
Do the case of noncompact finite-volume hyperbolic manifolds using the notes of Cohen-Sarnak, or Hejhal's book
adelic formula using Gelbart or Whitehouse
Examples on Hecke, Waldspurger, Watson-Ichino, and Ichino-Ikeda.
Reference: [I],[II],[J1],[J2]
This seminar is organized by Jiaqi Hou, Yu LUO and Simon Marshall. This page is took from Brian Lawrence's reading seminar