# Analytic theory of GL(3) automorphic forms and applications

November 17 to November 21, 2008

at the

American Institute of Mathematics, San Jose, California

organized by

Henryk Iwaniec, Philippe Michel, and K. Soundararajan

## Original Announcement

This workshop has the goal of providing a description of GL3 automorphic forms and their L-functions amenable to analytic number theorists and to explain the various approaches available to perform harmonic analysis on these spaces. A second objective will be to discuss the extension of some of the important tools existing in the GL2 theory to the GL3 context: a typical example is Kuznetzov's formula. A third objective will be to list some important problems known for GL2 and to identify the main obstructions to the extension of these to GL3: typical problems are non-vanishing problems for central values of L-functions and subconvexity problem. To achieve these goals we plan to bring together analytic number theorists and specialists from the theory of automorphic forms and related fields who are interested in analytic questions.

In addition to introductory lectures, the workshop will be centered around various "practical activities" conducted by different, possibly non-disjoint teams of people: the goal will be to study some specific problem of interest, possibly including

• Develop usable tools to perform averages in families of GL3 automorphic forms and in families GL3 L-functions in the various possible aspects: the goal will be to make as explicit as possible the spectral decomposition of the space of automorphic forms. A possibility would be to develop a workable form of the analog of the Kuznetzov formula in the GL3 context.
• Investigate the behaviour of the automorphic forms as one of the parameters attached to them vary.
• Prove non-vanishing results for GL3-L-functions in various aspects hopefully by using the mollification method; that should not be intrinsically difficult, the goal would be to get people used to the combinatorics underlying the Hecke algebra for GL3.
• Try to understand the difficulty underlying more serious problems like the subconvexity problem in its various aspects.

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Levels of distribution and the Affine Sieve
by  Alex Kontorovich,  Ann. Fac. Sci. Toulouse Math. (6) 23 (2014), no. 5, 933-966  MR3294598
Applications of the Kuznetsov formula on GL(3)
by  Valentin Blomer,  Invent. Math. 194 (2013), no. 3, 673-729  MR3127065
On the GL(3) Kuznetsov formula with applications to symmetry types of families of L-functions
by  Dorian Goldfeld and Alex Kontorovich,  Automorphic representations and $L$-functions, 263-310, Tata Inst. Fundam. Res. Stud. Math., 22, Tata Inst. Fund. Res., Mumbai, 2013  MR3156855
The subconvexity problem for $\mathrmGL_2$
by  Philippe Michel and Akshay Venkatesh