The Future of Trace Formulas (14w5001)
The theory of endoscopy, recently completed by the proof of the
Fundamental Lemma by Ngô and the endoscopic classification of automorphic
representations on classical groups by Arthur, can be regarded as a high
point in the development and applications of the Arthur-Selberg trace
formula to the Langlands program.
The goal of this workshop is to examine possible future directions for the
development of trace formulas. Among many recent developments and the new
directions that they offer, we contend to list the following:
1. "Beyond Endoscopy"
Langlands has proposed an ambitious plan for proving functoriality in
general by using the stable trace formula to study poles of L-functions.
It is a particularly apt time to have various aspects of this program
presented together, including the works of Venkatesh, Eddie Herman, Altug,
and Frenkel-Langlands-Ngô . All the works in the direction of "Beyond
Endoscopy" so far critically involve either the classical linear Poisson
summation formula, or conjectural non-linear versions of it – the latter
being crucial to Laurent Lafforgue's approach to functoriality to general
linear groups, via explicit construction of kernel functions for
functoriality. The relevance of generalized Poisson formulas to
Langlands functoriality is an important theme to explore.
2. Functoriality for spherical varieties and the relative trace formula
The relative trace formula has been tremendously successful in proving
instances of functoriality between spherical varieties and relations
between periods of automorphic forms. Most recently, a comparison proposed
by Jacquet and Rallis has been successfully carried out by Wei Zhang and
Zhiwei Yun to prove (under some conditions) the refinement of the unitary
Gan-Gross-Prasad conjecture due to Ichino-Ikeda and Neal Harris. With the
work of Sakellaridis-Venkatesh, the notion of "functoriality for spherical
varieties" has been rigorously formulated using the dual group of
Gaitsgory and Nadler, and generalizations of the Gross-Prasad and
Ichino-Ikeda conjectures have been proposed. The workshop will investigate
the relevance of the relative trace formula to these conjectures, and
various other aspects of the relative trace formula which are still under
3. Local trace formulas
The proof by Waldspurger and Moeglin-Waldspurger of the local Gross-Prasad
conjecture for orthogonal groups, and by Beuzart-Plessis for unitary
groups, uses essentially, a local twisted trace formula -- a twisted
analog of the classical local trace formula of Arthur.
A local relative trace formula of a different type has been developed in
the thesis of Feigon. The workshop will investigate such trace formulas
and the generalized character relations that they can prove; other methods
of proving local character relations, such as the ones used by Wei Zhang
for the unitary Gross-Prasad conjectures, will be compared to those.
The success of geometric methods in the proof of the Fundamental Lemma,
including analogs for the relative trace formula, have led to an effort of
understanding trace formulas over global function fields geometrically. In
this direction, Frenkel and Ngô have proposed a framework of
geometrization of both the geometric and spectral sides of trace formulas
over function fields, in the spirit of the geometric Langlands program.
Recent work of Ngô has also indicated the relevance of the geometrization
of trace formulas to problems in the setting of "Beyond Endoscopy" over
This workshop will serve as a focal point for researchers including a
number of new Ph.D.'s to interchange ideas and to further the development
of the subject.