BUNTES Fall 2024: Automorphic Representations
Time: Wed 11:00 AM - 12:00 PM
Location: CDS 365
Downloadable seminar information: Syllabus
References
- [G] Getz, J. R. An introduction to automorphic representations. Lecture notes. Available on Getz's website
- [GH] Getz, J. R., & Hahn, H. (2024). An Introduction to Automorphic Representations: With a view toward trace formulae. Graduate Texts in Mathematics vol. 300. Springer.
Other references include:
- Bump, D. (1997). Automorphic Forms and Representations. Cambridge: Cambridge University Press.
- Bushnell, C. J., & Henniart, G. (2006). The Local Langlands Conjecture for GL(2). Grundlehren der mathematischen Wissenschaften vol. 335. Springer Berlin Heidelberg.
- Jacquet, H., & Langlands, R. P. (1970). Automorphic Forms on GL(2): Part 1. LNM vol. 114. Springer Berlin Heidelberg.
- Brian Conrad's notes on adelic points: download
Schedule
| Week |
Date |
Topic |
Speaker |
| Week 1 |
Sep 11 |
Overview and affine group schemes |
Xinyu |
| Week 2 |
Sep 18 |
Automorphic representations |
Liqiang |
| Week 3 |
Sep 25 |
Non-archimedean representations |
Matt |
| Week 4 |
Oct 2 |
Archimedean representations |
Ashutosh |
| Week 5 |
Oct 9 |
Automorphic forms |
Xinyu |
| Week 6 |
Oct 16 |
Flath's theorem |
James |
| Week 7 |
Oct 23 |
Cuspidal and supercuspidal representations |
Zecheng |
| Week 8 |
Oct 30 |
Unramified representations |
|
| Week 9 |
Nov 6 |
Rankin-Selberg L-functions |
|
| Week 10 |
Nov 13 |
Simple trace formulas I |
|
| Week 11 |
Nov 20 |
Simple trace formulas II |
|
| Week 12 |
Nov 27 |
No Meeting: Thanksgiving |
|
| Week 13 |
Dec 4
|
Relative trace formulas |
|
| Week 14 |
Dec 11 |
TBD |
|
Descriptions of the topics
- Week 1: Overview and affine group schemes
- Summary of basic ideas and applications of automorphic representations. Review of affine group schemes and their adelic points.
- Week 2: Automorphic representations
- Hilbert space and unitary representations of locally compact groups. Haar measures. Gelfand-Pettis integrals.Discrete automorphic representations. Ref: [GH] Chapter 3, [G] Section 3.
- Week 3: Non-archimedean representations
- Smooth and admissible representations. Hecke algebras. Contragredients. Traces, characters, coefficients. Ref: [GH], Chapter 8, [G] Section 4
- Week 4: Archimdean representations
- Smooth and admissible representations. \((\mathfrak{g},K)\)-modules. Hecke algebras. Ref: Ref: [GH] Chapter 4, [G] Section 5
- Week 5: Automorphic forms
- Definitions of automorphic forms. Examples. From modular forms to automorphic forms. Casimir elements. Ref: {[GH]} Chapter 6, [G] Section 6
- Week 6: Flath's theorem
- Factorizations of automorphic representations: Flath's theorem and its proof. Ref: [GH] Chapter 5, [G] Sections 7,8.
- Week 7: Cuspidal and supercuspidal representations
- Definitions of cuspidal and supercuspidal representations. Jacquet module. Discreteness of the cuspidal spectrum. Ref: [GH] Chapters 8,9, [G] Section 11. (This may need a second talk)
- Week 8: Unramified representations
- Satake isomorphism. Ramanujan and Ramanujan-Petersson conjectures. Ref: [GH] Chapters 7, [G] Section 9.