July 18, 2024

Thematic Program on Automorphic Forms

Clay Mathematics Institute Summer School
on Harmonic Analysis, The Trace Formula and Shimura Varieties

June 2 - 27, 2003
held at The Fields Institute, Toronto


The Clay Mathematics Institute is organizing a summer school in automorphic forms in June, 2003. The school will be held at the Fields Institute in Toronto and will be aimed at graduate students and mathematicians within five years of their Ph.D.
The school will begin with three weeks of foundational courses centered around the trace formula: one course on the statement and proof of the trace formula, two courses providing background material on reductive groups and harmonic analysis on those groups, and a fourth course on Shimura varieties, which provide an illuminating application of the trace formula. The fourth week will consist of five short courses on more specialized topics related to the main themes of the school. While there are no formal prerequisites, preference will be given to applicants with some prior knowledge of algebraic groups or number theory.

Organizing Committee

James Arthur (Toronto), David Ellwood (Boston & CMI) , Robert Kottwitz (Chicago)

Summer School Lecture Courses: June 2 - June 20

  • June 2 - 20
    Introduction to the Trace Formula
    Instructor: James Arthur
The topic of this course will be the global trace formula for a reductive group over a number field. We shall begin with a brief overview of the subject, taking motivation from the case of compact quotient. We shall then prove as much of the general formula as we can. In the process, we shall introduce the orbital integrals and characters, and their weighted variants, that are the main terms in the trace formula. The deeper study of these local objects will be the subject of the course of Kottwitz, and the lectures of DeBacker and Hales in the final week. General applications of the trace formula will actually require two successive refinements, the invariant trace formula and the stable trace formula. If time permits, we shall discuss these refinements, and the local problems of comparison whose solutions are required for applications.
  • June 2- 20
    Introduction to Shimura Varieties
    Instructor: James Milne
  • Shimura varieties are the natural generalization of elliptic modular curves. Examples include the Hilbert modular varieties and the Siegel modular varieties. The fundamental theorem in the theory of Shimura varieties is the existence and uniqueness of canonical models over number fields. A primary goal of the course will be to obtain a good understanding of this theorem. In particular, we shall discuss the theorem of Shimura and Taniyama on complex multiplication, and the various ways of realizing Shimura varieties as moduli varieties. We expect also to include the following two topics: the structure of Shimura varieties modulo p, especially in the PEL case; boundaries of Shimura varieties and their various compactifications.

    Slides: Introduction to Shimura Varieties (.pdf) (.ps)

  • June 2-6
    Background from Algebraic Groups
    Instructor: Fiona Murnaghan
  • We will describe, without giving proofs, some of the main results in the theory of algebraic groups. The main emphasis will be on the classification and stucture of reductive algebraic groups.
  • June 9-20
    Harmonic Analysis on Reductive Groups and Lie Algebras
    Instructor: Robert Kottwitz
    This course will introduce the basic objects of study in harmonic analysis on reductive groups and Lie algebras over local fields: orbital integrals, their Fourier transforms (in the Lie algebra case) and characters of irreducible representations (in the group case). The emphasis will be on p-adic fields and the Lie algebra case (to which the group case can often be reduced using the exponential map). Some of the main theorems involving these objects will be discussed: Howe's finiteness theorem; Shalika germs, the local character expansion and its Lie algebra analog; local integrability of Fourier transforms of orbital integrals; and the Lie algebra analog of the local trace formula.

Advanced Short Courses: June 23- June 27
List of Lecturers:

An Introduction to Homogeneity with Applications
Stephen DeBacker
In the early 1990s, J.-L. Waldspurger established a very precise version of Howe's finiteness conjecture (for the Lie algebra). We shall discuss this result and some of its applications to harmonic analysis on reductive p-adic groups.
Geometry and Topology of Compactifications of Modular Varieties
Mark Goresky
We will describe the construction, basic properties and applications of the Baily-Borel (Satake) compactification, the Borel-Serre and reductive Borel-Serre compactifications, and the toroidal compactifications.
Bad Reduction of Shimura Varieties
Thomas Haines
We will survey recent work on the bad reduction of Shimura varieties. In particular, we will focus on the computation of the local Hasse-Weil zeta functions at bad primes, and connections with the Langlands correspondence.
An Introduction to the Fundamental Lemma
Tom Hales

A collection of conjectural identities between integrals on reductive groups has become known as the "Fundamental Lemma." These lectures will describe these conjectural identities, and will discuss the progress that has been made toward their proof.
Analytic Aspects of Automorphic Forms
Peter Sarnak

The analytic theory of L-functions has many applications to number theory and automorphic forms (and visa versa).We describe some of this theory and its applications. Specifically to the Ramanujan Conjectures and other spectral problems associated with quotients of homogeneous spaces and to arithmetical problems such as quadratic forms.


Graduate and Postdoctoral Funding

Funding is available to graduate students and postdoctoral fellows (within 5 years of their PHD) to attend the summer school. We anticipate that funding will be available for 90 graduate students and young mathematicians. Interested candidates must forward with their application, a letter of recommendation from their mathematical advisor or a senior mathematician. Standard support amounts will include funds for local expenses and accommodation plus economy travel.
Deadline for applications was February 15, 2003

Mathematical Preparation

There will be no formal prerequisites for the summer school. However, the three longer courses will all be presented from general standpoint of reductive algebraic groups. This is the reason for the initial one week course, devoted to the statement and description of some of the basic properties of algebraic groups. Since this course will not include proofs, participants are encouraged to come with some prior understanding of the subject. A familiarity with any of the topics in the following references would be a definite asset to bring to the summer school.

A. Borel, Linear Algebraic Groups, Benjamin, 1969.
J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, 1972.
T. Springer, Linear Algebraic Groups, Birkhäuser, 1981.
J.-P. Serre, Complex Semisimple Lie Algebras, Springer-Verlag, 1987.

Participants will be assumed to have some knowledge of number theory. The main theorems
of class field theory will be reviewed in the course on Shimura varieties, but again without proof. A good general reference is,
J. Cassels, and A. Fröhlich, Algebraic Number Theory, Thompson, 1967.
The thesis of Tate, reprinted in this volume, is especially recommended, for its introduction to adèles, and its construction of abelian L-functions.
A good introductory reference to the general theory of automorphic forms is the proceedings of the Edinburgh instructional conference:
T.N. Bailey and A.W. Knapp, Representation Theory and Automorphic Forms, Proc. Sympos. Pure Math., 61, Amer. Math. Soc., 1996.

The Clay Mathematics Institute

The Clay Mathematics Institute (CMI) is a private, non-profit foundation, dedicated to increase and to disseminate mathematical knowledge.
The primary objectives and purposes of The Clay Mathematics Institute (CMI) are, to increase and disseminate mathematical knowledge, to educate mathematicians and other scientists about new discoveries in the field of mathematics,to encourage gifted students to pursue mathematical careers, and to recognize extraordinary achievements and advances in mathematical research.

Applications to Attend

The final date to register was February 15, 2003, therefore we are no longer accepting applications.

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