# The stable trace formula, automorphic forms, and Galois representations (08w5040)

Arriving in Banff, Alberta Sunday, August 17 and departing Friday August 22, 2008

## Organizers

(University of Toronto)

Michael Harris (Universite de Paris 7)

Eric Urban (Institut Mathématique de Jussieu)

Vinayak Vatsal (Mathematics Department, University of British Columbia)

## Objectives

A generation of number theorists has grown up studying the theory
of elliptic modular forms. The arithmetic, geometric, and analytic
techniques required to understand automorphic forms on $GL(2)$
have become widely familiar, and can be learned from a variety of
excellent texts. In contrast, the theory of automorphic forms on other
groups, and specifically the arithmetic theory of automorphic
forms on Shimura varieties, remains the province of a small
number of specialists. This, we believe, is less due to the intrinsic
difficulty of the subject than to the fact that it has developed primarily as
a dialogue among experts. Moreover, with the exception of the volume
The zeta functions of Picard modular surfaces" published by the CRM in
Montreal in 1992, which treats in depth the case of the group $U(3)$, articles
on the stable trace formula have invariably treated the subject in
complete generality. This undoubtedly contributed to the rapid development
of the field, but made the literature much less accessible to colleagues with
backgrounds in number theory or arithmetic geometry rather than Lie theory.

It has long been apparent that a full understanding of elliptic modular
forms requires the development of an arithmetic theory of automorphic
forms on groups other than $GL(2)$. The recent work of Skinner
and Urban, among others, on generalizations of the Iwasawa Main
Conjecture, and the recent proof by Clozel, Harris, Shepherd-Barron,
and Taylor of the Sato-Tate Conjecture for elliptic curves over
$Bbb{Q}$ with non-integral $j$-invariant, have shown that this
is not only necessary but is also possible, even with techniques
presently available. The primary goal of the proposed workshop
is to contribute to creating a situation where number theorists
will be able to make use of the most recent developments in the
theory of automorphic forms on higher-dimensional groups with
no less ease than they have hitherto done with the $GL(2)$-theory.
We expect that the forthcoming books of Arthur and of the Paris
automorphic forms seminar, emphasizing the specific cases of
orthogonal and symplectic groups and unitary groups, respectively,
will help provide a transition between
the $GL(2)$-theory and the general theory.

The Paris project expects to produce four books, devoted, respectively,
to I. The stable trace formula and the Fundamental
Lemma ; II. Zeta functions of Shimura varieties ; III. $p$-adic families
of automorphic forms and Galois representations; and IV. Arithmetic
applications. The first three books should concentrate essentially
on automorphic forms on unitary groups and the related theory of
general linear groups. Their provisional tables of
contents are appended below. Volume IV, on the other
hand, remains almost completely open. The second goal of the
proposed workshop is to encourage number theorists to mine the
developed theory for applications. One obvious line of applications is to use
deformations of non-tempered cohomology classes to construct infinite
subgroups of Selmer groups, as in the work of Skinner-Urban and
Bella"i{}che-Chenevier; indeed Chenevier has already spoken in the Paris
seminar on his work with Bella"i{}che generalizing their previous
work to arbitrary dimension, conditional on the instances of the Arthur
multiplicity conjectures expected to be proved in the first two volumes.
Other possible applications are to the Arakelov theory of Shimura
varieties and relations to derivatives of $L$-functions, as in Kudla's
program; the construction of Euler systems in the cohomology of
Shimura varieties; or ergodic properties of special cycles on Shimura
varieties, as considered from various points of view by Vatsal and Cornut,
Clozel-Ullmo, and Jiang-Li-Zhang.

Moreover, we expect the theory of the stable trace formula
to make considerable progress between now
and the time of the workshop. Ng^o and others have been
developing a program to prove the endoscopic Fundamental
Lemma for groups other than $U(n)$, using geometric methods
related to the Hitchin fibration. Arthur's book will undoubtedly
stimulate work on twisted versions of the Fundamental Lemma;
the first steps have been taken by Waldspurger, who has reduced
the Fundamental Lemma for twisted endoscopy to what he calls standard
and non-standard Fundamental Lemmas in which no twisting is
apparent. Finally, the full strength of Arthur's stabilization of the
invariant trace formula requires a variant of the Fundamental Lemma
for weighted orbital integrals, which requires a completely new set of
geometric insights; this is the subject of work in progress of Chaudouard
and Laumon.

With regard to the main application of the Paris project's first three books --
the construction of Galois representations attached to cohomological
automorphic representations of GL(n) over CM fields, under a duality hypothesis --
considerable progress has been made during the past two years. Under a series
of simplifying hypotheses that suffice for the construction of Galois representations,
a simple version of the stable trace formula for unitary groups now seems within
reach, using only the version of the Fundamental Lemma proved by
Laumon-Ng^o; this is the subject of work in progress of Moeglin and Labesse.
As for the analysis of the zeta functions of unitary Shimura varieties,
S. W. Shin, in his thesis work under the direction of Richard Taylor at
Harvard, is in the process of developing a synthesis of the work of
Kottwitz and Harris-Taylor which will allow him to apply the techniques
of the latter in situations where endoscopy is non-trivial
(or alternatively, to apply Kottwitz' techniques in situations of
bad reduction). Harris has sketched a strategy
for obtaining all the expected Galois representations from these results, using
p-adic continuity in the setting of eigenvarieties of higher dimension constructed by Chenevier.

For these reasons we propose a two-part workshop, to be divided into
two weeks. The first week would be similar to a summer school and
would address the first two goals mentioned above. It would be directed at
graduate students working in the field of automorphic forms as well as at
number theorists wishing to gain familiarity with the techniques and results
afforded by the stable trace formula. The contents would be based on the
forthcoming books of Arthur and the first two volumes of the Paris automorphic
forms seminar, versions of which should exist by the summer of 2008, when
the workshop should take place. The second week would be a research
conference on questions related to the contents of the books. Progress in
the stabilization of the trace formula and the twisted trace formula will
feature prominently, but we also hope to present new results on the
construction of $p$-adic families of automorphic forms and related
Galois representations, as in Volume III of the Paris seminar, and
perhaps even some arithmetic applications as promised in Volume IV.