RTF reading seminar Fall 2024 - Spring 2025

SYNOPSIS

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.

Schedule

References

• [B] Daniel Bump Automorphic Forms and Representations
• [C] Cogdell, Lectures on L-functions, Converse Theorems, and Functoriality for \(\mathrm{GL}_n\)
• [T] Chao Li Topics in Automorphic Forms by Jack Thorne
• [G] Stephen S. Gelbart Automorphic Forms on Adele Groups
• [GH1] Jayce R. Getz and Heekyoung Hahn An Introduction to Automorphic Representations
• [GH2] Goldfeld and Hundley, Automorphic Representations and L-Functions for the General Linear Group Volume 1
• [F] Yuval Z. Flicker - On distinguished representations
• [Z] Yihang Zhu - Introduction to the Langlands Program (summer school at MCM and PKU)

• [A] James Arthur - An Introduction to the Trace Formula
• [CS] P. Cohen, P. Sarnak, Selberg trace formula Chapter 6, Chapter 7
• Brian Conrad, Seminar on Jacquet-Langlands;
• [W] David Whitehouse, An introduction to the trace formula;
• [H] Dennis A. Hejhal, The Selberg Trace Formula for PSL (2,R), Volume 1, Volume 2;
• [S] Atle Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, Scaned file.
• [Sn] Andrew Snowden, The Jacquet-Langlands Correspondence for GL(2).

• [I] Ichino - Trilinear forms and the central values of triple product L-functions
• [II] Ichino-Ikeda - On the Periods of Automorphic Forms on Special Orthogonal Groups and the Gross–Prasad Conjecture
• [J1] Jacquet, Hervé - Sur un résultat de Waldspurger
• [J2] Jacquet, Hervé - Sur un resultat de Waldspurger II

• [J1] Jacquet, Hervé - Sur un résultat de Waldspurger
• [J2] Jacquet, Hervé - Sur un resultat de Waldspurger II

• Chao Li's notes, Relative trace formulae and the Gan-Gross-Prasad conjectures

Outline

Part 0: Introduction

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 \(G=GL_n \times GL_m\).

Part 1: Overview of Automorphic representations

• Week 3: Automorphic representations for \(\mathrm{GL}_2\) - [B] Sections 3.3, 3.4

  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

• Week 4: Representation of \(\mathrm{GL}_2(\mathbb{R})\) - [B] Sections 2.4.2.5.2.7 or [GH2] Ch7

  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)

• Week 5: Representations of \(\mathrm{GL}_2(\mathbb{Q}_p)\) - [B] or [GH2] Ch 6

  Definition of smooth, admissible representations

  Parabolic induction, principal series, Supercuspidal, and Jacquet module

  If time permitted, Satake isomorphism [GH1] 7.1, 7.2

• Week 6 and 7: L-groups, L-functions - [GH1] ch 7

  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

Part 2: Regular trace formula and Jacquet-Langlands

• Trace formula for compact quotient.

  Present it using Selberg's original paper

• Trace formula for non-compact quotient.

  Do the case of noncompact finite-volume hyperbolic manifolds using the notes of Cohen-Sarnak, or Hejhal's book

• Trace formula for adelic quotient.

  adelic formula using Gelbart or Whitehouse

• Jacquet-Langlands correspondence

Part 3: Introduction to period integral formulas and applications.

• Overview of those period integral formulas, and mention some application

  Examples on Hecke, Waldspurger, Watson-Ichino, and Ichino-Ikeda.

• Philosophy behind those formulas

  Reference: [I],[II],[J1],[J2]

Part 4: Jacquet’s proof of Waldspurger formula

• Waldspurger formula
• RTFs for Waldspurger period and Hecke period
• Comparison of RTFs and the proof of Waldspurger formula

Part 5: Further topics

• Introduction to Gan–Gross–Prasad conjecture
• Jacquet-Rallis approach to the GGP conjecture
• Beuzart-Plessis's proof of Jacquet-Rallis fundamental lemma

This seminar is organized by Jiaqi Hou, Yu LUO and Simon Marshall. This page is took from Brian Lawrence's reading seminar