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BUNTES Fall 2024: Automorphic Representations


Time: Wed 11:00 AM - 12:00 PM
Location: CDS 365

Downloadable seminar information: Syllabus

References

  1. [G] Getz, J. R. An introduction to automorphic representations. Lecture notes. Available on Getz's website
  2. [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.
  3. Notes on simple trace formulas from Conrad's seminar: Trace formulas I, Trace formulas II, and Trace formulas III. Also link to the website of the seminar.

Other references include:

  1. Bump, D. (1997). Automorphic Forms and Representations. Cambridge: Cambridge University Press.
  2. Bushnell, C. J., & Henniart, G. (2006). The Local Langlands Conjecture for GL(2). Grundlehren der mathematischen Wissenschaften vol. 335. Springer Berlin Heidelberg.
  3. Jacquet, H., & Langlands, R. P. (1970). Automorphic Forms on GL(2): Part 1. LNM vol. 114. Springer Berlin Heidelberg.
  4. 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 Ashutosh
Week 9 Nov 6 Rankin-Selberg L-functions James
Week 10 Nov 13 Simple trace formulas I Liqiang
Week 11 Nov 20 Simple trace formulas II Jacksyn
Week 12 Nov 27 No Meeting: Thanksgiving
Week 13 Dec 4
 Relative trace formulas Zecheng
Week 14 Dec 11 TBD Matt

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.

Week 9: Rankin-Selberg L-functions
Local and global Rankin-Selberg L-functions. Multiplicity one and strong multiplicity one. Ref: [GH] Chapter 11, [G] Section 10.

Week 10: Simple trace formulas I
Motivations of trace formulas. Trace kernel. Orbital integrals. Statement of simple trace formulas. Proof (of a simple case) if time permits. Ref: [GH] Chapters 16, 17, 18, [G] Section 12. Notes from Conrad's seminar: Trace formulas I, Trace formulas II, and Trace formulas III

Week 11: Simple trace formulas II
Applications. Proofs of more general cases of simple trace formulas if time permits. Ref: [GH] Chapters 18, 19, [G] Sections 12, 13. Notes from Conrad's seminar: Trace formulas I, Trace formulas II, and Trace formulas III