**Topics in automorphic forms**

This year's topic is the
*Langlands
correspondence for general reductive groups over function fields*. We will study Vincent Lafforgue's
fundamental paper *Chtoucas pour les groupes
réductifs et paramétrisation de Langlands globale*, which attaches a
semisimple Langlands parameter to every cuspidal automorphic representation of
a reductive group G over the field of functions of a smooth projective algebraic
curve X over a finite field. If
time permits, we will also consider Lafforgue's work with Alain Genestier,
which uses the global construction to define a local Langlands parametrization.

Assume
G is a split reductive group; then a Langlands parameter for G is a
homomorphism from the arithmetic fundamental group of X (minus a finite set of
points) to the Langlands dual group . Lafforgue
breaks with the method used by Drinfel'd and Laurent Lafforgue to prove the
global Langlands correspondence for GL(n), which (like the method pioneered by
Langlands and Kottwitz to study the zeta functions of Shimura varieties) is
based on a comparison of the Arthur-Selberg trace formula with the
Grothendieck-Lefschetz trace formula.
Instead, Lafforgue constructs combinatorial data, which we call -*pseudocharacters*, that contain enough information to define
semisimple representations, by defining a new algebra of operators, called *excursion
operators*, that incorporate the action of the global Galois group on the
moduli spaces (stacks) of *shtukas with paws*. The
excursion operators generalize the familiar Hecke correspondences, and
Lafforgue's main theorem is a decomposition of the space of cusp forms (of
fixed level) over characters of the algebra of excursion operators and the
identification of these characters with global Langlands parameters.

The
bulk of Lafforgue's paper is concerned with a close analysis of the geometry of
these moduli stacks, specifically with showing how the geometric Satake
correspondence interprets the action of the unramified Hecke correspondences in
such a way as to identify the restriction of the global Langlands parameter at
an unramified place with what is predicted by the arithmetic Satake correspondence. In so doing, Lafforgue introduces
methods of geometric representation theory into arithmetic in an utterly novel
way. The scope of these new
methods promises to be much broader than the applications to the global
Langlands correspondence that motivated Lafforgue's work. For example, Peter Scholze has outlined
a program to interpret some of Lafforgue's constructions in the setting of
perfectoid geometry, and to apply them to the local Langlands program for
p-adic groups.

Lafforgue's
paper is available at the arXiv link indicated above; there are also shorter
versions (in English as well as French) on Lafforgue's home page. The course is provisionally scheduled
to meet on Tuesdays and Thursdays from 1:10-2:25, but this may change, and the
scheduling will be flexible to accomodate travel schedules of the instructor as
well as students. Dates on the
tentative schedule below in black correspond to the official schedule. Dates in red
through Mid-February are for background in automorphic forms, to be held at
6:30 PM on Wednesdays, in a room to be determined. Subsequent dates in red will be discussed later.

**Tentative
weekly schedule**

Week 1, January 26, 27, 28 Statement of main results of Vincent Lafforgue

(a) Introduction to the results of Vincent Lafforgue

(b) automorphic representations, L-group, Hecke algebra and classical Satake isomorphism

(c) moduli stacks of G-shtukas, paws, excursion operators, pseudocharacters

Week 2, February 2, 3, 4 Theory of pseudocharacters, following Richardson and Lafforgue

(a) Definition of pseudo-representations

(b) Automorphic representations, cusp forms, finiteness properties

(c) Relation between pseudocharacters and representations,
introduction to Hecke stacks

** References:** G. Harder, Chevalley
Groups Over Function Fields and Automorphic Forms, *Annals of Math.* 1974

A. Borel, Automorphic forms on reductive groups, informal notes

Week 3, February 9 (**no class February 11**) Moduli of G-torsors

(a) Cusp forms and moduli stack
of shtukas with no paws

**References:** C. Sorger, Lectures
on moduli of principal G-bundles over algebraic curves

Week 4, February 16, 17, 18 Moduli of G-shtukas

(a) Moduli stacks of vector bundles and G-shtukas

(b)
*to be
determined*

(c) Moduli stacks of G-shtukas and Hecke correspondences

Week 5, February 23, 25 Loop groups and affine Grassmannians

(a) Affine Grassmannians as ind-schemes, the case of GL(n)

(b) Affine Grassmannians in general

**References:** U. Görtz, Affine
Springer fibers and affine Deligne-Lusztig varieties

Week 6, March 1, 3 Geometric Satake isomorphism

(a) Beilinson-Drinfeld affine Grassmannian, geometric Satake isomorphism, 1

(b) Geometric Satake isomorphism, 2

**References:** T. Richarz, A new approach to
the geometric Satake equivalence

There will be no class on March 8 and 10th, but on March 11, W. Zhang's talk in the automorphic forms seminar will be closely related to the topics of this course.

Week 7, March 22, 24 Uniformization 2

(a) Geometric Satake isomorphism, 3; local structure of moduli spaces of shtukas

(March 23 there will be a book talk at Book Culture, on a topic unrelated to this course.)

(b) Coalescence, creation, and annihilation operators

Week 8, March 29, 30, 31 Excursion operators

(a) Definitions of excursion operators, 1

(b) Perverse
sheaves [NOTE CHANGE!], with a
proof of semisimplicity of the Satake category

(c) Properties of excursion operators

**References**
(for (b))**:** G. Williamson, An
illustrated guide to perverse sheaves

Week 9, April 5, 7 Drinfeld's lemma and applications

(a) Hecke correspondences

(b) Partial Frobenius operators and end of the definition of excursion operators

Week 10-11, April 12, 21 Langlands parametrization [NO CLASS 4/14 or 4/19!]

(a) Potential automorphy, 1

(b) Potential automorphy, 2

Week 12, April 26, 28, Compatibility with geometric Satake

(a) Compatibility with geometric Satake (minuscule case)

(b)
Compatibility with geometric Satake (general case)

** **

**Some references**

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