Families of Automorphic Forms and the Trace Formula

Welcome!

We would like to welcome all participants of this Simons Symposium to take place January 26 – February 1, 2014 at the St. Regis Bahia Beach Resort in Rio Grande, Puerto Rico.

This endeavor of the Simons Foundation offers a great opportunity for researchers to study families of automorphic representations of higher rank groups. The symposium will investigate the trace formula, plancherel equidistribution, p-adic families and other recent techniques from harmonic analysis and representation theory. One goal is to try set an agenda for the field.

We are working to create a meeting week that provides ample time for discussion and active collaboration. There will be significant time devoted to discussing open problems. The participants are encouraged to suggest in advance specific topics they would like to see addressed.

Report

Here is a detailed report on the activities that took place during the symposium.

Schedule

Monday Tuesday Wednesday Thursday Friday
Symposium
Program
9-9:15
opening
9-9:50
Gee
9-9:50
Hoffman
9-9:50
Speh
9-9:50
Kim
9:30-10:30
Calegari

9:50-10:40
open discussion
Calegari
9:50-10:40
open discussion
Urban
9:50-10:40
open discussion
Gee
9:50-10:40
problem session
Hida
break
Symposium
Program
11:10-12:00
Muller
11:10-12:00
Miller
11:10-12:00
Muller
11:10-12:00
Nadler
11:10-12:00
Holowinsky


lunch
break


Symposium
Program
3-3:50
Hida
3-3:50
Soundararajan
3-4
Shin
3-4
Hoffman
Miller
3:50-4:40
Templier


3:50-4:40
problem session
organizers
4-5
open discussion
Gordon
3:50-4:40
open discussion
Shin
break early dinner
Symposium
Program
5:30-6:00
problem session
Miller
5:10-6:00
Urban
5:10-6:00
Kaletha
4:30-6:00
Nadler

dinner

7:00-10:45

excursion!

* * * * * * *
Symposium
Program
8-8:30
problem session
8-8:30
problem session
8-8:30
problem session
8:40-9:30
Gordon
8:40-9:30
Cluckers
8:40-9:30
Marshall

In addition to regular talks the program features open discussions lead by a moderator and interactive problem sessions.

Here are some pictures of the kayak excursion on Wednesday.

This Symposium is made possible by the generous support of the Simons Foundation. It is one of four Simons Symposia in 2014. Further details on the Simons Symposia are available here.

group picture

Notes / Links / References / Open problems

Steve Miller is taking tex notes of the talks, you can see it here
Calegari
"Families of automorphic forms of cohomological type"
Consider a reductive group G over Q. What can one say about the average behavior of automorphic forms for G of cohomological type? When G(R) has discrete series, one can say quite a lot using the trace formula. However, For a general group G, the cohomological forms form a very "thin" subset of all forms, and so are harder to study using analytic methods. I will discuss various conjectural predictions for what one might expect in the general case, as well as give various theoretical approaches to these problems.
Casselman
"Steps towards a Schwartz space for reductive monoids".
This talk will be about the relationship between analysis on p-adic monoids, Langlands' L-functions, and perverse sheaves, following recent suggestions of Ngô and work of Li Wenwei, generalizing the classical results of Tamagawa and Godement-Jacquet concerning Schwartz functions on matrix algebras.
Hida
Growth of Hecke fields over a slope 0 family
Here are the slides of the talk. Here is a slide pdf file of a list of problems on the growth of Hecke fields over a slope 0 analytic families (and related topics). After an explanation of notation/terminology, the problems are in Section 8-13 (pages 8-13).
Templier
Families of L-functions and their symmetry
See this preprint.
Gordon and Cluckers
Gordon: "Using model theory to obtain uniform bounds for orbital integrals"
Cluckers: "Motivic integration and transfer principles"
See the following survey article, which should serve as an outline or abstract of the talks.
Gee
I will discuss conjectures of Buzzard and Gouvea about the p-adic variation of Hecke eigenvalues of mod p modular forms as the weights and levels vary, together with the relationship of these kinds of problems to automorphy lifting theorems, and a new analogy between automorphy lifting theorems and the trace formula.
Here are some notes for the discussion session.
Muller
Limit multiplicities for principal congruence subgroups of GL(n)
see this article.
Soundararajan
I will speak on recent work with Maksym Radziwill on upper bounds for moments of L-functions. One application is to calculate upper bounds for the usual size of Tate-Shafarevich groups of rank zero quadratic twists of elliptic curves.
Urban
Eigenvarieties
Hoffman
The trace formula and prehomogeneous vector spaces
I will speak on an approach to express the coefficients of weighted orbital integrals in the trace formula in terms of prehomogeneous zeta integrals. It has been realised here for the symplectic group of rank 2, and there is a preprint on algebraic prerequisites in general.
Here are the slides of the talk.
Miller
Please follow this link for a summary by S. Miller of works on $L$-functions and low-lying zeros. You will find slides of talks, introductory and survey articles as well as recent research.
Here are slides for the problem session.
Shin
Sato-Tate conjecture for families, see this preprint.
Here are notes for the open discussion.
Speh
Symmetry breaking and the Gross-Prasad conjectures for real orthogonal groups
Here are the slides of the talk.
Nadler
Affine character sheaves
Kaletha
"Rigid inner forms and endoscopy"
Here are the slides of the talk. The refined local Langlands conjecture predicts a bijective correspondence between refined parameters and individual representations (rather than packets) of a connected reductive group. This refinement is needed for many applications involving the stable trace formula, a basic example being the multiplicity formula for discrete automorphic representations. We will discuss a precise statement of this conjecture for an arbitrary connected reductive group G defined over a local field F of characteristic zero. This statement is based on the construction of a certain Galois gerb (in the sense of Langlands-Rapoport) over F, whose arithmetic properties lead to a normalization of the endoscopic objects involved in the local Langlands conjecture.
Marshall
Cohomology growth on U(3)
Kim
Holowinsky
Subconvexity bounds for Rankin-Selberg L-functions. See this and that paper.

To make the Symposium more interactive we invite all participants to upload any kind of notes / links / references / open problems that you think if of interest. This material will be available above to all the participants. (just send it to me by email at [templier.nicolas at zzgmailzz] in any format, I'll take care of uploading it here on the webpage)

* Particularly helpful will be some background material about your talk. Such as abstract, or recommended survey and research articles, pointers to good references, links to resources available online. Anything that you think can be helpful for the other participants to be aware of in relation to your talk.

* We invite you to submit problems in advance to share with the participants. You may do so in writing (tex, pdf or raw) prior to the conference so we can make a compilation and distribute it at the symposium.

As a source of inspiration, here are some examples of problems sessions from a previous symposium on tropical geometry. Also here are links to problems from a conference in honor of W. Schmid and from a conference in honor of B.Gross. In summary any material you want to share with the other participants prior the meeting will be very welcome and will certainly help towards making the symposium high level and productive.

Participants

Proceedings

The proceedings will be published within a series of Simons symposia published by Springer. (see here for the proceedings from last year). It will contain both research articles and survey articles.

Practical

A lot of information can be found on the registration webpage, including hotel, travel, and contacts.

Talks. The conference room is equipped with 3 blackboards. These are double-sided blackboards. We also have audio-visual support with a projector and white screen if you are using slides in your talk. (a technician will be available to help but you probably would only need to hook your laptop via a vga cable and for macbooks via a VGA adapter). There will be a microphone. The talks probably won't be recorded.