SOUTHERN CALIFORNIA
NUMBER THEORY DAY
U.C. SAN DIEGO, MAY 9, 2009
Abstracts:
(1) Sug Woo Shin (Chicago)
Title: Plancherel density theorem for automorphic
representations
Abstract:
Let S be a finite set of finite primes. Let G be a connected
reductive group over Q such that G(R) has a discrete series.
Following an introduction to the problem, I prove that the
Scomponents of discrete automorphic representations of G(A)
are equidistributed with respect to the Plancherel measure on
the unitary dual of G(Q_S). The result is built upon work of
Serre, Sauvageot and many others. One immediate corollary is
an existence theorem for automorphic representations.
(2) Matthew Emerton (Northwestern)
Title: padically completed cohomology
and the padic Langlands program
Abstract: Speaking at a general level, a major
goal of the padic Langlands program (from a global, rather than local, perspective)
is to find a padic generalization of the notion of automorphic eigenform,
the hope being that every padic global Galois representation will
correspond to such an object. (Recall that only those Galois representations
that are motivic, i.e. that come from geometry, are expected to correspond
to classical automorphic eigenforms.)
In certain contexts (namely, when one has Shimura varieties at hand),
one can begin with a geometric definition of automorphic forms, and generalize
it to obtain a geometric definition of padic automorphic forms. However, in
the nonShimura variety context, such an approach is not available. Furthermore,
this approach is somewhat remote from the representationtheoretic point of view
on automorphic forms, which plays such an important role in the classical Langlands
program.
In this talk I will explain a different, and very general, approach
to the problem of padic interpolation, via the theory of padically
completed cohomology. This approach has close ties to the padic
and mod p representation theory of padic groups, and to noncommutative
Iwasawa theory.
After introducing the basic objects (namely, the padically completed cohomology
spaces attached to a given reductive group), I will explain
several key conjectures that we expect to hold, including the conjectural
relationship to Galois deformation spaces. Although these conjectures
seem out of reach at present in general, some progress has been made towards
them in particular cases. I will describe some of this progress, and along
the way will introduce some of the tools that we have developed for studying
padically completed cohomology, the most important of these being the Poincare
duality spectral sequence.
