# Workshop on L-functions and Trace Formula

May 11-13, 2015 at Purdue University. A working workshop, aimed at reporting the latest progress on the subject.

## Organizers

- William Casselman, University of British Columbia
- Freydoon Shahidi, Purdue University

## Schedule

Workshop Schedule. All talks will take place in Beering Hall at Purdue University.## Proposed Talks

**Ali Altug, Columbia University: Trace formula beyond endoscopy and analytic number theory. **

I will talk about trace formula and analytic number theory, and how one may combine the two to get more explicit forms of the trace formula. I'll mainly discuss the elliptic part and how one can control the arithmetic issues as well as the analytic ones. I'll also discuss some applications. All this is aimed at questions motivated by beyond endoscopy.

**James Arthur, University of Toronto: Problems beyond endoscopy. **

I shall speak on further questions related to my talk last year at Banff. These include the problems relating to the geometic contributions of spectral terms in what I called the stabilization of the (hypothetical) r-trace formula. Of particular interest are the nontempered spectral terms for (say) quasisplit orthgonal and sympletic groups. My talk would concern questions rather than answers!

**Bill Casselman, University of British Columbia: Computing basic functions**

There are formulas for p-adic basic functions due to W-w Li, Ngo, and Sakellaridis, but they don't give a very explicit idea of what they look like. It turns out to be feasible and illuminating to compute examples for groups of very low rank. I'll exhibit some, explain the difficulties in dealing with groups of high rank, and speculate about asymptotic behaviour.

**Jayce Getz, Duke University: Remarks on a paper of Frenkel, Langlands and Ngo.**

**Tobias Finis, Free University of Berlin, Germany: On the Continuity of the Geometric Side of Arthur's Trace Formula. **

I'll report on joint work with Erez Lapid that extends the methods of our 2011 Compositio paper to deal at least with the coarse geometric expansion of the (non-invariant) trace formula.

**Tasho Kaletha, Princeton University: On the automorphic spectrum of non-quasi-split groups. **

The Langlands-Arthur conjectures provide a description of the discrete automorphic representations of connected reductive groups defined over global fields, as well as of the irreducible admissible representations of such groups defined over local fields. When the group in question is quasi-split, a precise form of these conjectures has been known for a long time and important special cases have recently been proved. For non-quasi-split groups (such as special linear, symplectic, and special orthogonal groups over division algebras), the conjectures have been vague and their proof out of reach.* * In this talk we will present a precise formulation of the local and global conjectures for arbitrary connected reductive groups in characteristic zero. It is based on the construction of certain Galois gerbes defined over local and global fields and the study of their cohomology. These cohomological results place the conjectures for classical groups well within reach of the currently available methods.

**Wen-Wei Li, Chinese Academy of Science, Beijing: Remarks on zeta integrals and their functional equations.**

**Jasmin Matz, Mathematical Institute, University of Bonn, Germany: An introduction to beyond endoscopy. **

*I could give an overview of Langlands' "Beyond Endoscopy". So my tentative plan would be to give a summary of part I & II of Langlands BE and then maybe elaborate on some of the points mentioned there - this depends a little on what the other participants plan to speak about. Two central points are probably to explain the problems of removing the contribution of the trivial representation on the geometric side of the trace formula, and of isolating the representations of Ramanujan type. However, I have not worked out any details yet.*

**Bao Chau Ngo, University of Chicago: ****Digression on formal arcs spaces and Basic function and singularities in the formal arc space of reductive monoid**

**Yiannis Sakellaridis, Rutgers-Newark and NTU Athens: TBD**

**Freydon Shahidi, Purdue University:** *I plan to pose certain questions in connection with Ngo's monoid construction of L-functions as motivated by my own work on comparison of root numbers and related matters, as well as questions cocerning non-vanishing of certain periods in terms of Arthur parameters and possible connections to the work of Sakellaridis-Venkatesh. ** *