Beyond Endoscopy

Beyond endoscopy and geometric terms

James Arthur
University of Toronto
October 1, 2016
Beyond Endoscopy is the proposal by Langlands for using the trace formula to attack the Principle of Functoriality. Many of the problems at this (early) stage concern the geometric terms in the trace formula. Most immediate perhaps is to extend Altug's application of Poisson summation from $\mathrm{GL}(2)$ to more general groups. One could then try to identify the contribution of the nontempered representations on the spectral side to the resulting Fourier transforms of geometric terms.

Asymptotics for Hecke eigenvalues with improved error term

Jasmin Matz
Universität Leipzig
October 1, 2016
Asymptotics for the distribution of Hecke eigenvalues in families of automorphic forms are useful to study families of L-functions, provided one has a sufficiently good estimate for the error term of the asymptotic. I want to present ongoing joint work with T. Finis in which we give an upper bound for the error term with an explicit dependence on the Hecke operator. For certain applications an improvement of this bound would be necessary, but to go beyond our bound it seems necessary to remove part of the geometric side of the trace formula.

L-functions, monoids and Bessel functions

Freydoon Shahidi
Purdue University
October 1, 2016
This is a survey of how the existing theories of L-functions are in agreement with Braverman-Kazhdan/Ngo's construction of L-functions which generalizes that of Godement-Jacquet. One hopes that such insights may play a role in understanding the corresponding Fourier transform and Poisson formula which are of particular interest in functoriality.

Regular supercuspidal representations

Tasho Kaletha
University of Michigan
October 1, 2016
Jiu-Kang Yu has given a general construction of supercuspidal representations of tamely ramified reductive p-adic groups. We will show that most of these representations can be parameterized by conjugacy classes of pairs consisting of an elliptic maximal torus and a character of it, subject to a simple and explicit root-theoretic condition. We will then draw a remarkable parallel between the characters of these representations and the characters of discrete series representations of real reductive groups.

Decomposing symmetric powers

Bill Casselman
University of British Columbia
September 30, 2016
One problem raised recently by Langlands in connection with the trace formula was the decomposition of symmetric powers of irreducible representations of $\mathrm{GL}(2)$. There is a classical formula for this. I shall explain a new version of this formula and, with luck, describe computer experiments examining the extent to which the idea will work for other groups.

Geometric side of the trace formula and related problems

Ali Altuğ
Massachusetts Institute of Technology
September 30, 2016
A general problem in Beyond Endoscopy is to understand the terms on the geometric side of the trace formula and to get enough control on their Fourier (or related) transforms. The problem itself has several different ingredients with different flavors (some local some global, for instance) like the singularities of orbital integrals, irregular variation of class numbers of tori in families, contributions of non-tempered representations etc.