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

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

## Organizers

James Arthur (University of Toronto)

Michael Harris (Universite de Paris 7)

Eric Urban (Institut MathÃ©matique 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.

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.