Math 263A: Topics in Representation Theory

 Fall 2024

TuTh 10:30 AM - 11:50 AM, Room 381T.
In this topic course, we will discuss function field arithmetic, mainly on automorphic functions over function fields, and geometric represention theory over formal discs and finite fields, including cases with ramifications. We explain how to study arithmetic invariants via different methods, based on moduli spaces of G-bundles, special cycles, perverse sheaves and Langlands program. Note in positive characteristic, some methods in char 0 e.g. complex Hodge theory, differential equations and hyperbolic geometry are not available.
There are some fundamental principles: Weil's intepretation of automorphic forms as functions on moduli of vector bundles, Grothendieck's function-sheaf dictionary, analogs for Riemann surfaces, geometrization of spherical functions (Satake isomorphisms), categorical Langlands program and cateogrical traces.

Instructor: Zhiyu Zhang.

Office Hour: by appointment.

Exercises:

Lecture notes : under construction. See Lecture notes of a similar course offfed by Z. Yun in Stanford, Winter 2015.

Topics:

  • L-functions (Yun-Zhang, Abdurrahman-Venkatesh).
  • Higher special cycles, theta correspondences and derived Fourier analysis (Feng-Yun-Zhang).
  • L-sheaves (Ben-Zvi-Sakellaridis-Venkatesh).
  • Geometric class field theory with ramifications (Lang-Serre-Rosenlicht, Deligne, Guignard, Takeuchi, Campbell-Hayash..).
  • Automorphic functions as the trace of Frobenius (or how to invent shtukas) (Gaitsgory, [AGKRRV20a],[AGKRRV20b], [AGKRRV21]..).
  • Ramanujan conjectures (Sawin-Templier).
  • L-functions and geometric Langlands over the projective line (Gross, V. Lafforgue).
  • Tate's thesis.
  • Springer theory, affine Springer fiber and orbital integrals.
  • Relative orbital integrals, germ expansions and fundamental lemmas.
    Lecture Schedule
  • Lecture 1, Sep 24.
  • Lecture 2, Sep 26.
  • Lecture 3, Oct 1. No class.
  • Lecture 4, Oct 3. No class.
  • Lecture 5, Oct 8.
  • Lecture 6, Oct 10.
  • Lecture 7, Oct 15.
  • Lecture 8, Oct 17.
  • Lecture 9, Oct 22. Zoom class.
  • Lecture 10, Oct 24. Zoom class.
  • Lecture 11, Oct 29.
  • Lecture 12, Oct 31.
  • Lecture 13, Nov 5. No class (Democracy Day).
  • Lecture 14, Nov 7.
  • Lecture 15, Nov 12.
  • Lecture 16, Nov 14.
  • Lecture 17, Nov 19.
  • Lecture 18, Nov 21.
  • Lecture 19, Nov 26. No class (Thanksgiving Recess).
  • Lecture 20, Nov 28. No class (Thanksgiving Recess).
  • Lecture 21, Dec 3.
  • Lecture 22, Dec 5.

    Additional topics (Time permitted) :

  • 0-dimensional components of moduli spaces of local systems, monodromy and automorphic representations (Yun, Jakob-Yun, Fargeman).
  • General groups e.g. exceptional groups (Heinloth-Ngo-Yun, Yun, Bockle-Feng-Harris-Khare-Thorne).
  • Dimension of spaces of automorphic forms and Sturm-type bounds (Armana-Wei).
  • Hurwitz stacks and Selmer stacks (Ellenberg-Venkatesh-Westerland, Ellenberg-Landesman).
  • Plectic conjectures (Nekovar-Scholl, Tamiozzo, Daniel Li-Huerta).
    Additional references:
  • Stanford Topic Course on Shtukas, Z. Yun, Winter 2015.
  • Talk on Spectral decomposition , Vincent Lafforgue.
  • An arithmetic application of geometric Langlands, S. Raskin.
  • Whittaker patterns in the geometry of moduli spaces of bundles on curves, E. Frenkel, D. Gaitsgory, and K. Vilonen.
  • Section 9 and 11, Relative Langlands Duality, D. Ben-Zvi, Y. Sakellaridis, A. Venkatesh.
  • Symmetric varieties for endoscopic groups, S. Leslie.
  • Lectures on Shtukas, C. Xue, E. Viehman, Summer School on the Arithmetic of the Langlands Program, 2023.
  • Lectures on Local Systems in Algebraic-Arithmetic Geometry, H. Esnault.
  • Torsion volume forms, F. Naef, P. Safronov.