# Relative trace formula and periods of automorphic forms

August 24 to August 28, 2009

at the

American Institute of Mathematics, San Jose, California

organized by

Herve Jacquet, Erez Lapid, and Akshay Venkatesh

## Original Announcement

This workshop will be devoted to the study of the relative trace formula and periods of automorphic forms.

In particular, we hope to formulate a precise general conjecture for the exact value of period integrals which encompasses all known cases (either proven, e.g. torus periods on GL(2) (Waldspurger), unitary periods on the general linear group (Jacquet), or conjectural e.g. the work of Ichino and Ikeda on the Gross-Prasad period). The relative trace formula relates periods integrals on two different groups, and often reduces a "difficult" period integral to an "easy" one, thus providing a powerful tool to attack the putative conjecture. Thus far the study of the RTF has been primarily example-based, and we hope to (begin to) develop a general theory.

The main topics for the workshop are

• Examine all known cases of period integrals, with an eye towards formulating a general conjecture
• Formulation of a "general" relative trace formula and its "compatibility" with the general conjecture.
• Methodology of the relative trace formula: e.g. adapting the techniques of Ngo to relative setting, development of a local relative trace formula.
• Examine the compatibility of the general conjecture with "explicit constructions" e.g. backward lifting, Theta.
• Periods in the case where there is not multiplicity one (e.g. unitary and orthogonal periods on the general linear group).

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups
by  Wee Teck Gan, Benedict H. Gross, and Dipendra Prasad
Restrictions of representations of classical groups: examples
by  Wee Teck Gan, Benedict H. Gross, and Dipendra Prasad
Transfer to characteristic zero: appendix to "Fundamental Lemma of Jacquet-Rallis in positive characteristics" by Zhiwei Yun
by  Julia Gordon
The fundamental lemma of Jacquet-Rallis in positive characteristics
by  Zhiwei Yun