Topics on Automorphic L-functions I
This course provides an introduction to the theory of automorphic L-functions.
The first part develops the harmonic analysis of GL(1) following Tate’s thesis, covering local and global fields, locally compact abelian groups and their Pontryagin duals, Schwartz–Bruhat function spaces, Fourier transforms, and zeta integrals together with their absolute convergence, functional equations, and meromorphic continuation.
The second part focuses on the Godement–Jacquet theory, which generalizes Tate’s approach to the standard L-functions of automorphic representations of
GL(n) for arbitrary n.
The first part develops the harmonic analysis of GL(1) following Tate’s thesis, covering local and global fields, locally compact abelian groups and their Pontryagin duals, Schwartz–Bruhat function spaces, Fourier transforms, and zeta integrals together with their absolute convergence, functional equations, and meromorphic continuation.
The second part focuses on the Godement–Jacquet theory, which generalizes Tate’s approach to the standard L-functions of automorphic representations of
GL(n) for arbitrary n.
Lecturer
Date
18th September, 2026 ~ 22nd January, 2027
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Friday | 13:30 - 16:05 | Shuangqing | ZOOM 02 | 518 868 7656 | BIMSA |
Prerequisite
The course is designed for graduate students interested in pursuing research in the Langlands Program. Prerequisites include familiarity with linear and abstract algebra, real and complex analysis, basic Fourier analysis, basic point-set topology, and basic representation theory of Lie groups and p-adic groups.
Syllabus
Week 1: Introduction to L-function theory in the Langlands Program.
Week 2: Locally compact abelian groups(LCA), Haar measures, and Pontryagin Duality.
Week 3: Tate's thesis: Local Theory.
Week 4: Tate's thesis: Global Theory.
Week 5: Godement-Jacquet Theory, an overview.
Week 6: Local Theory: Convergence Lemmas.
Week 7: Local Theory: Induced representations.
Week 8: Local Theory: Reduction to absolutely cuspidal case.
Week 9: Local Theory: Division Algebras.
Week 10: Local Theory: Absolute cuspidal representations.
Week 11: Local Theory: Spherical functions and Spherical Representations.
Week 12: Local Theory: Archimedean case.
Week 13: Local Theory: Unitary representations.
Week 14: Global Theory: Automorphic forms.
Week 15: Global Theory: Convergence Lemmas.
Week 16: Global Theory: Zeta integrals and the main theorems.
Week 2: Locally compact abelian groups(LCA), Haar measures, and Pontryagin Duality.
Week 3: Tate's thesis: Local Theory.
Week 4: Tate's thesis: Global Theory.
Week 5: Godement-Jacquet Theory, an overview.
Week 6: Local Theory: Convergence Lemmas.
Week 7: Local Theory: Induced representations.
Week 8: Local Theory: Reduction to absolutely cuspidal case.
Week 9: Local Theory: Division Algebras.
Week 10: Local Theory: Absolute cuspidal representations.
Week 11: Local Theory: Spherical functions and Spherical Representations.
Week 12: Local Theory: Archimedean case.
Week 13: Local Theory: Unitary representations.
Week 14: Global Theory: Automorphic forms.
Week 15: Global Theory: Convergence Lemmas.
Week 16: Global Theory: Zeta integrals and the main theorems.
Reference
1. J. Tate, Fourier analysis in number fields and Hecke’s zeta‑functions. In J. W. S. Cassels \& A. Fröhlich (Eds.), Algebraic Number Theory (Proceedings of an instructional conference organized by the London Mathematical Society, pp. 305–347). Academic Press, 1967.
2. R. Godement, H. Jacquet, Zeta Functions of Simple Algebras (Lecture Notes in Mathematics, Vol. 260). Springer‑Verlag, 1972.
3. J. Cogdell, Notes on L-functions of GL(n). Lecture notes given at the School of automorphic forms on GL(n), Trieste, 2000.
2. R. Godement, H. Jacquet, Zeta Functions of Simple Algebras (Lecture Notes in Mathematics, Vol. 260). Springer‑Verlag, 1972.
3. J. Cogdell, Notes on L-functions of GL(n). Lecture notes given at the School of automorphic forms on GL(n), Trieste, 2000.
Audience
Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
Yes
Language
English