Number theory learning seminar 2017-2018

The seminar will meet Wednesdays 1:30--3:30pm in Room 384H. This year's seminar will focus on the Converse Theorem for GL(2) over global fields, roughly following notes of Godement's IAS seminar on the topic that takes a geodesic path through the parts of Jacquet-Langlands' GL(2) book to make a beeline to that result (setting aside other things done in there). After one has seen the motivation from a representation-theoretic interpretation of the classical Weil-Hecke converse theorem for GL(2) over the rational field (in the language of classical cuspidal modular forms), there are three main aspects: local representation theory (non-archimedean and archimedean), global adelic representation theory, and analytic properties of global L-functions. As part of this development, it is essential that we work with Hilbert-space representations in order to access crucial tools from functional analysis (such as compact operators).

In the fall (after an initial motivational lecture) we will first discuss reduction theory for connected reductive groups over a general global field; in the end we just need this for GL(2), but even that for a general global field involves passing away from the GL(2)-case via Weil restriction to reduce to a possibly non-split group. With that out of the way, we take up some facts about the cuspidal part of L2 for a general reductive group over a global field, and then finally focus on GL(2) for the rest of the seminar, more-or-less following Godement with supplementary references that arise along the way.

Familiarity with functional analysis on Hilbert spaces, adelic class field theory, classical modular forms, Tate's thesis, and the basic structure of algebraic groups will be assumed (though for many purposes you can focus on GL(2) at the cost of some loss of conceptual clarity on the algebraic group aspects).

Here are some references relevant to this year's seminar (in approximate order of appearance):

[M] Modular forms, book by Miyake
[Sp] Reduction theory over global fields, Springer
[B] Introduction aux groupes arithmetiques, book by Borel
[T] Modular forms and automorphic representations , notes by Trotabas
[Ge] Automorphic forms on adele groups, book by Gelbart
[Bu1] Notes on Representations of GL(r) Over a Finite Field, notes by Bump
[Bu2] Automorphic forms and representations, book by Bump
[N] Automorphic forms on GL(2) , U.Chicago course notes by Ngo (disclaimer: these notes have not been revised or edited in any way since they were first prepared)
[Ga] Decomposition and estimates for cuspforms, notes by Garrett
[Go] Notes on Jacquet-Langlands' theory, IAS lecture notes by Godement
[Sn] A. Snowden's notes on Jacquet-Langlands



Notes -- use at your own risk.

These are informal notes. They may change without warning.

Fall quarter
1 Oct. 4 Conrad Motivation from the classical converse theorem [M, 4.3, 4.5] .pdf
2 Oct. 11 Conrad Reduction theory I ([Sp], [B]) .pdf
3 Oct. 18 Conrad Reduction theory II ([Sp], [B]) .pdf
4 Oct. 25 Conrad Reduction theory III ([Sp], [B]) .pdf
5 Nov. 1 Love, Sherman Adelization of classical modular forms I ([T], [Ge], [N]) .pdf
6 Nov. 8 Dhillon Kirillov model ([Go, 1.1-1.4], [Bu1], [Bu2, 4.4]) .pdf
7 Nov. 15 Howe Discreteness for cuspidal L2 for reductive groups I ([Go, 3.1], [Ga]) .pdf
8 Nov. 30 Howe Discreteness for cuspidal L2 for reductive groups II ([Go, 3.1], [Ga])
9 Dec. 6 Love, Sherman Adelization of classical modular forms II ([T], [Ge])
10 Dec. 13 Tsai Admissible representations and supercuspidals I [Go, 1.5-1.7]
Winter quarter
11 Jan. 10 Tsai Admissible representations and supercuspidals II [Go, 1.5-1.7]
12 Jan. 17 Rosengarten Principal series [Go, 1.8-1.11]
13 Jan. 24 Zaman Local functional equation [Go, 1.12-1.15]
14 Jan. 31 Devadas, Zavyalov Spherical and unitary classification I [Go, 1.16-1.20]
15 Feb. 7 Devadas, Zavyalov Spherical and unitary classification II [Go, 1.16-1.20]
16 Feb. 14 Tam Archimedean constructions ([Go, 2.1-2.2], [Sn, 6-7])
17 Feb. 21 Landesman Irreducible components ([Go, 2.3-2.4], [Sn, 6-7])
18 Feb. 28 Tsai Archimedean Kirillov model ([Go, 2.5], [Sn, 6-7])
March 7 Cancelled (Arizona Winter School)
March 14 Cancelled (MSRI Hot Topics)
Spring quarter
19 April 4 Thorner Archimedean functional equation and local L and ε factors [Go, 2.6-2.8]
20 April 11 Masullo Adelic representations and global Hecke algebras [Go, 3.2-3.3],
21 April 18 Masullo Global Whittaker model and multiplicity one [Go, 3.4-3.5]
22 April 25 Dore Global automorphic L-function and functional equation [Go, 3.6-3.7]
23 May 2 Feng Relation with classical L-functions and classical multiplicity one ([T], [Ge])
24 May 9 Howe Converse Theorem [Go, 3.8], applications in classical case (time permitting)
25 May 16 TBD Overflow
26 May 23 TBD Overflow
27 May 30 TBD Overflow
28 June 6 TBD Overflow