Welcome to the Learning Seminar on Automorphic Forms. This seminar aims to provide an introduction to automorphic forms, continuing on the seminar last semester. We'll follow the books by Bump (Automorphic Forms and Representations) and Goldfeld (Automorphic Forms and L-Functions for the Group GL(n,R)) . We'll also have some guest lectures from other universities on more specialised topics like: Sieve methods for Automorphic forms and Hida Families.
Organisers: Aditya Ghosh, Alan Zhao, Austin Lei
Date: TBD
Time: TBD
Location: TBD
| Serial No. | Date | Speaker | Abstract | References | Notes |
|---|---|---|---|---|---|
| 1 | 27-Jan | Austin Lei | Maass forms and Whittaker functions for SL(n, Z): We will discuss Maass forms and their Fourier expansions for SL(n, Z). We will also discuss the construction of the Jacquet Whittaker function and their functional equation. | [2] Chapter 5 | notes |
| 2 | 3-Feb | Alan Zhao | Vogan diagrams for real semisimple Lie groups: The existence of a Cartan involution distinguishes compact and non-compact Lie groups simply yet precisely. Whereas these adjectives are usually placed as assumptions at the outset, the classification of real Lie groups depends on how they intertwine, i.e. the Cayley transform. This talk will further realize these details. | - | |
| 3 | 10-Feb | Austin Lei | Maass Forms and L-functions for SL(n, Z): We will discuss the properties of the Fourier expansion for SL(n, Z) Maass forms, Hecke operators, and the Godement-Jacquet L-function. Time permitting, we will specialize to SL(3, Z) Maass forms. | [2] Chapter 9 | notes |
| 4 | 17-Feb | Aditya Ghosh | Classification of irreducible (g,K)-modules and the spectral problem: The talk will cover some basic representation theory of Lie Groups. We will also classify irreducible (g,K) - modules for GL(2,R) and briefly discuss their unitracity. Finally we will discuss how these occur in the in decomposition of L^(\Gamma \ G). We will also discuss how this is relevant when considering automorphic forms for congruence subgroups and how it connects to the adelic viewpoint. | [1] Chapter 2.5 - 2.6 | |
| 5 | 24-Feb | Austin Lei | Maass Forms and L-functions for SL(3, Z): We will review the theory of SL(n, Z) Maass forms and specialize to SL(3, Z). We will discuss of the proof of the multiplicity one theorem for SL(3, Z) Whittaker functions, the functional equation for the Godement-Jacquet L-function, and the double Dirichlet series. Time permitting, we also begin discussion of Eisenstein series. | [2] Chapter 6 | notes |
| 6 | 3-Mar | Alan Zhao | Doi-Naganuma Lifting and Rankin Selberg: This talk will provide an introduction to base change, a special case of Langlands functoriality. We work in the classical setting and prove a result of Doi and Naganuma. In the GL(2) case, we will need to understand modular forms over algebraic number fields. We’ll see what the form that adelic representation theory takes for such objects. | - | |
| 7 | 10-Mar | - | Seminar cancelled. | - | |
| 8 | 17-Mar | - | Spring Break, no seminar. | - | |
| 9 | 24-Mar | Austin Lei | Langlands Eisenstein Series: Langlands Eisenstein series: We will talk about the construction of Langlands Eisenstein series and their Fourier expansions and functional equations. We will also discuss applications of Langlands Eisenstein series towards the functional equation of Maass forms and the nonvanishing of L-functions on the line Re(s) = 1. | [2] Chapter 10 | notes |
| 10 | 31-Mar | Aditya Ghosh | Classification of (g,K)-modules and unitracity: The talk will cover (g,K)-modules and classify them upto infintesimal equivalence. We will also look at which of these give unitary representations. | [1] Chapter 2.7 - 2.9 | |
| 11 | 7-Apr | Aditya Ghosh | Automorphic forms and representations over GL(2) and Whittaker Models: The talk will define automorphic forms and representations over GL(n). We will also define Whittaker Models, locally and globally and how it connects to the theory of automorphic forms. | [1] | |
| 12 | 14-Apr | - | Talk cancelled. | - | |
| 13 | 21-Apr | Ajmain Yamin | Selberg Trace Formula | TBD | |
| 14 | 28-Apr | Vishal Muthuvel | TBD | TBD | |
| 15 | 5-May | - | Organizers out of town, no seminar. | - |
Aditya Ghosh: ag4794 (at) columbia (dot) edu
Alan Zhao: asz2115 (at) columbia (dot) edu
Austin Lei: ayl2158 (at) columbia (dot) edu