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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.
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