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 L^{2} 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)
[BH] The Local Langlands Correspondence for GL(2), book by Bushnell and Henniart
[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
Fall quarter | ||||
1 | Oct. 4 | Conrad | Motivation from the classical converse theorem [M, 4.3, 4.5] | |
2 | Oct. 11 | Conrad | Reduction theory I ([Sp], [B]) | |
3 | Oct. 18 | Conrad | Reduction theory II ([Sp], [B]) | |
4 | Oct. 25 | Conrad | Reduction theory III ([Sp], [B]) | |
5 | Nov. 1 | Love, Sherman | Adelization of classical modular forms I ([T], [Ge], [N]) | |
6 | Nov. 8 | Dhillon | Kirillov model ([Go, 1.1-1.4], [Bu1], [Bu2, 4.4]) | |
7 | Nov. 15 | Howe | Discreteness for cuspidal L^{2} for reductive groups I ([Go, 3.1], [Ga]) | |
8 | Nov. 30 | Howe | Discreteness for cuspidal L^{2} 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 | Rosengarten | Supercuspidals and parabolic induction [BH, Section 10, 11.5] | |
14 | Jan. 31 | Zaman | Local functional equation [Go, 1.12-1.15] | |
15 | Feb. 7 | Devadas, Zavyalov | Spherical and unitary classification I [Go, 1.16-1.20] | |
16 | Feb. 14 | Devadas, Zavyalov | Spherical and unitary classification II [Go, 1.16-1.20] | |
17 | Feb. 21 | Tam | Archimedean constructions ([Go, 2.1-2.2], [Sn, 6-7]) | |
18 | Feb. 28 | Landesman | Irreducible components ([Go, 2.3-2.4], [Sn, 6-7]) | |
March 7 | Cancelled (Arizona Winter School) | |||
19 | March 14 | Tsai | Archimedean Kirillov model ([Go, 2.5], [Sn, 6-7]) | |
Spring quarter | ||||
20 | April 4 | Thorner | Archimedean functional equation and local L and ε factors [Go, 2.6-2.8] | |
21 | April 11 | Masullo | Adelic representations and global Hecke algebras [Go, 3.2-3.3], | |
22 | April 18 | Masullo | Global Whittaker model and multiplicity one [Go, 3.4-3.5] | |
23 | April 25 | Dore | Global automorphic L-function and functional equation [Go, 3.6-3.7] | |
24 | May 2 | Feng | Relation with classical L-functions and classical multiplicity one ([T], [Ge]) | |
25 | May 9 | Howe | Converse Theorem [Go, 3.8], applications in classical case (time permitting) | |
26 | May 16 | TBD | Overflow | |
27 | May 23 | TBD | Overflow | |
28 | May 30 | TBD | Overflow | |
29 | June 6 | TBD | Overflow |