THEMATIC PROGRAMS

March 29, 2024

January-June 2012 Thematic Program on Galois Representations

Coxeter Lecture Series
February 29, March 1 & 2,2012
Michael Harris
Université Paris 7 (Jussieu)
Open Questions about Motives Attached to Automorphic Forms

February 29 - General Lecture
Fields Institute, Room 230
- 3:30 p.m.,

The proof by Ngô Báo Châu of the Fundamental Lemma has led to confirmation of an important prediction of the Langlands program, namely the existence of a correspondence between certain kinds of representations of Galois groups of number fields and certain classes of automorphic representations. Combined with the methods introduced by Wiles, this correspondence has been applied to solve traditional problems in algebraic number theory, including the Sato-Tate conjecture. The lecture will review some of these results and situate them in the general framework of the Langlands program.

March 1 & 2 - Specialized Lectures
Fields Institute, Room 230
- 3:30 p.m

The Galois representations attached to an automorphic representation are in most cases realized on the l-adic cohomology of a Shimura variety. Other cohomology theories give rise to different kinds of arithmetic structures, and each such structure can be interpreted as a realization of the motive attached to the automorphic representation. Relations among Galois representations are expected to reflect relations among the corresponding motives, which in turn imply explicit relations among integrals attached to automorphic forms on different groups, a vast generalization of Shimura’s theory of CM periods for arithmetic holomorphic automorphic forms.

I will outline some of the motivating conjectures and will describe a few of them in detail, especially those connected to conjectures of Ichino and Ikeda on special values of L-functions.


Michael Harris is an internationally renowned expert in the theories of automorphic forms, Shimura varieties, and Galois representations, and his research has ranged over a wide range of topics related to these fields of investigation.

In some of Harris’s earliest work he introduced Iwasawa-theoretic techniques in the context of non-abelian p-adic Lie groups, techniques which are now very topical due to the widespread interest in non-commutative Iwasawa theory and the p-adic Langlands program. In other early work, he initiated the study of automorphic vector bundles on Shimura varieties, including the study of their canonical models and their cohomology, thus opening up an important technique for the study of the arithmetic of automorphic forms on general Shimura varieties. He applied this technique, and others, to make a detailed study of L-functions attached to automorphic forms in a range of contexts, verifying various rationality conjectures of Deligne in many situations.

Together with Richard Taylor, in 1999 he proved the local Langlands conjecture for GLn, and also constructed n-dimensional global Galois representations attached to self-dual cuspforms on GLn over totally real and CM fields. Building on this work came a series of papers, joint with Taylor and other collaborators (Clozel, Shepherd-Barron, Barnet-Lamb, and Geraghty), which served to establish the Sato–Tate conjecture for modular forms, and more generally initiated a framework for studying in the context of n-dimensional Galois representations problems which had previously been approachable only in the more classical setting of two-dimensional representations. In part as a means of encouraging number-theorists to take advantage of this new framework, Harris led the so-called “Paris book project”, a series of volumes dedicated to explaining aspects of theory of automorphic forms on unitary groups, including the stable trace formula, the proof by Laumon and Ngo of the fundamental lemma for unitary groups, and functoriality between unitary groups and GLn with the goal of explaining to number theorists the results that are available in the n-dimensional context.

Harris’s research achievements have earned him numerous prizes and honours, including being an invited speaker at the 2002 ICM in Beijing, winning the Grand Prix Scientifique de la Fondation Simone, and the Clay Research Award, shared with Richard Taylor, in 2007.

Speakers in the Coxeter Lecture Series (CLS) have made outstanding contributions to their field of mathematics. The CLS consists of a series of three one-hour lectures.

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