# Titles and Abstracts

## Ali Altug, Boston University

**Title:** Functoriality and related problems

**Abstract:** The functoriality conjectures, since their introduction by Langlands in 1967, have been among the most central problems in modern number theory and the theory of automorphic forms. In the fifty years they have been around they proved to be notoriously dicult to study. Although there has been a lot of progress, and many cases of functoriality are proved, the general conjectures (even for *GL(2)*) are wide open. Relatively recently, Langlands has suggested a method, now known as “Beyond Endoscopy”, to attack the functoriality conjectures in their full generality. The strategy proposes to bring the trace formula and analytic number theory in a novel way to study automorphic *L-*functions.

In this talk I will begin with describing beyond endoscopy and discussing the underlying analytic problems. I will then talk about certain problems about the trace formula arising from the considera- tions in beyond endoscopy, and recent suggestions of Arthur regarding the structure of the geometric side of the trace formula. If time permits, I will also add a few words about various other problems inspired by beyond endoscopy.

## Charlotte Chan, University of Michigan

**Title:** Affine Deligne--Lusztig varieties at infinite level

**Abstract:** Affine Deligne--Lusztig varieties are related to the reduction of certain Shimura varieties. I will construct an affine Deligne--Lusztig variety at infinite level for *GL*_{n} and its pure inner forms, and describe its cohomology groups. This object is closely related to the semi-infinite Deligne--Lusztig variety, which is the *p*-adic analogue of classical Deligne--Lusztig theory proposed by Lusztig in 1979. It will turn out that the torus eigenspaces of the cohomology are (essentially) concentrated in a single degree (which can be explicated in terms of Howe factorizations) and the relevant varieties are maximal in the sense that the number of rational points attains its Weil--Deligne bound. This is joint work with Alexander Ivanov.

## Jim Cogdell, Ohio State University

**Title:** Functoriality and integral representations for GSpin

**Abstract:** In a pair of papers in 2006 and 2014, Asgari and Shahidi showed the existence of a functorial lift from globally generic cuspidal representations π of GSpin groups to *GL*_{n} . In the second paper, utilizing the descent from *GL*_{n} to GSpin of Hundley and Sayag, they characterized the image representations Π on *GL*_{n} . Along the way, they needed a relation between the poles of the twisted L-function L(s, π × τ ), with τ a cuspidal representation of some GLm, and an Eisenstein series on a GSpin group induced from τ. This is precisely the type of relation one expects from a Rankin-Selberg type integral representation for L(s,π×τ). Following the lead of Ginzburg, Rallis, and Soudry, we would like to explain our thoughts on these integral representations. This talk will be part survey of the past results of Asgari and Shahidi (functoriality) and part an explanation of work in progress with Asgari and Shahidi (integral representations). (Slides)

## Ellen Eischen, University of Oregon

**Title:** p-adic L-functions and Eisenstein series on unitary groups

**Abstract:** I will discuss a construction of p-adic L-functions, with a focus on p-adic L- functions attached to cuspidal automorphic representations of unitary groups. I will highlight how this construction relates to more familiar ones of Serre, Katz, and Hida, and I will emphasize the role of properties of certain automorphic forms (analogous to the role played by modular forms in their work). This is joint work with Michael Harris, Jian-Shu Li, and Christopher Skinner.

## Roger Howe, Texas A&M University

**Title:** On the rank of representations of classical groups

**Abstract:** The representation theory of reductive groups over locally compact fields has been a subject of intensive study for around 75 years, and much progress has been made. However, there is still much work to be done before the subject can be considered well understood. Even the representation theory of groups over finite fields, despite the remarkable classification theory of Lusztig, presents questions that are not readily answered with current understanding.

The bulk of reductive groups are, or are closely related to, the family of groups known as ``classical groups". This talk will focus on the notion of rank of a representation of a classical group. It turns out that rank provides a means of describing the representations that are relatively ``small" in various senses. In particular, for groups over finite fields, it provides a way of systematically describing the representations of small dimension. As an added benefit, it seems to be closely related to some issues of harmonic analysis. (Joint work with Shamgar Gurevich. )

## Båo Châu Ngô, University of Chicago

**Title:** Geometry of some generalized affine Springer fibers

## Abhishek Parab, Purdue University

**Title:** Absolute convergence of the Twisted Arthur-Selberg Trace Formula

**Abstract:** We show that the distributions occurring in the geometric and spectral side of the twisted Arthur-Selberg trace formula extend to non-compactly supported test functions. The geometric assertion is modulo a hypothesis on root systems proven in many useful cases including when the group is split. It extends the work of Finis-Lapid (and Muller, spectral side) in the non-twisted setting. In the end, we will give an application towards residues of Rankin-Selberg L-functions. (Slides)

## A. Raghuram, IISER Pune and Purdue University

**Title:** On the special values of "certain *L*-functions."

**Abstract:** This talk will start with an introduction to a circle of ideas that concerns the cohomology of arithmetic groups and its relation to the special values of certain Langlands-Shahidi *L*-functions. I will begin by introducing the general context in which one can study the notion of Eisenstein cohomology. I will then explain some results of Harder on the cohomology of the boundary of the Borel-Serre compactification of a locally symmetric space and its relation with induced representations of the ambient reductive group. Once this context is in place one may then try to view Langlands's constant term theorem, which sees the ratios of products of automorphic L-functions, in terms of maps in cohomology. Whenever this is possible one is able to prove rationality results for ratios of critical values of certain automorphic *L*-functions. I will briefly discuss some results, obtained partly in collaboration with Harder and partly with Chandrasheel Bhagwat, on the special values of various Rankin-Selberg L-functions. (Slides)

## Chen Wan, Institute for Advanced Studies

**Title:** The local trace formula for the Ginzburg-Rallis model and the generalized Shalika model

**Abstract:** We will first discuss a local trace formula for the Ginzburg-Rallis model. This trace formula allows us to prove a multiplicity formula for the Ginzburg-Rallis model, which implies that the summation of the multiplicities on every tempered Vogan L-packet is always equal to 1. Then we will talk about an analogy of this trace formula for the generalized Shalika model, which implies that the multiplicity for the generalized Shalika model is a constant on every discrete Vogan L-packet.

Representation Theory Conference