## Final Report Summary - ANERAUTOHI (Analytic and ergodic aspects of automorphic forms on higher rank groups)

This project was designed to study a wide range of analytic aspects of higher rank groups, especially their L-functions and their applications. The initial step was obviously to master the objects themselves, which was achieved as follows. Firstly, a weekly workshop on the subject of this project was set up in Zürich. Then, Emmanuel Kowalski and Guillaume Ricotta participated to several events closely related to the theme of this project. For instance,

. The Ohio State University (February 3-8, 2012);

. Théorie des Nombres et Applications (CIRM, Luminy, January 16-20, 2012);

. Rencontre de théorie analytique et élémentaire des nombres (IHP, Paris, June 11, 2012).

These preliminary efforts enable us to reach the following main results during this project.

Very recently, Etienne Fouvry (Université Paris-Sud), Emmanuel Kowalski and Philippe Michel (EPFL Lausanne) published online an impressive series of preprints on the subject of trace functions. Very briefly speaking, they used the involved machinery of l-adic sheaves to get very concrete results in analytic number theory, such as problems concerning the asymptotic distribution of arithmetic functions in residue classes, which are very classical in analytic number theory and have been considered from many different points of view.

It seems natural to explore the same type of statistical questions for higher divisor functions, or for Fourier coefficients of automorphic forms on higher-rank groups, especially because of the philosophy which relates the distribution properties of primes in arithmetic progressions with

that of higher divisor functions.

Based on the works mentioned just before, Emmanuel Kowalski and Guillaume Ricotta show that the multiple divisor functions of integers in invertible residue classes modulo a prime number, as well as the Fourier coecients of GL(N) Maaβ cusp forms for all N > 2, satisfy a central limit theorem in a suitable range. Such universal Gaussian behaviour relies on a deep equidistribution result of products of hyper-Kloosterman sums. Note that there are currently very few statements of analytic number theory, which are known to hold for cusp forms on GL(N) for arbitrary N. The present work adds a further property to a very short list. It is submitted to Geometric and Functional Analysis. It turns out that this joint work implies many side effects, which are currently investigated by Guillaume Ricotta.

In a joint work with Ivan Fesenko (University of Nottingham) and Masatoshi Suzuki (The University of Tokyo) published in Annales de l'Institut Fourier, Guillaume Ricotta determined a necessary and sufficient condition under which the expected analytic properties of the L-function of an elliptic curve over a number fiels hold. Even if this is weaker than the automorphic property of the L-function, this problem remains deep.

A significant part of Guillaume Ricotta's research work with Emmanuel Royer (Université Blaise Pascal, Clermont-Ferrand, France) was about zeros of symmetric power L-functions. They found the asymptotic behaviour of several statistics associated to the zeros of these families of L-functions, for example the n-level density. This lead to one publication in Forum Mathematicum.

In a joint work in progress with Roman Holowinsky (Ohio State University) and Emmanuel Royer, Guillaume Ricotta is interested in the asymptotic behaviour of the supremum norm of GL(3) automorphic forms as the spectral parameter gets large. Guillaume Ricotta was invited to visit the Ohio State University twice (February 3-18, 2012 and April 21-May 4, 2013) in order to think intensively on this problem. They hope to achieve this work in a short term run.

Emmanuel Kowalski and Guillaume Ricotta gave many seminar presentations and conference lectures on the results described above. For example,

. The Ohio State University Number Theory Seminar (February 9, 2012);

. The Ohio State University Number Theory Seminar (April 23, 2013);

. Théorie Analytique des Nombres (CIRM, Luminy, June 19, 2013).

Last but not least, Guillaume Ricotta supervises two doctoral students. This project was of much benefits to them since their research problems are closely related to the main theme of it. One of them, Damien Bernard, did a statistical study of the first non-trivial zero of families of L-functions attached to automorphic forms of higher-rank groups. His work is submitted. He will defend his dissertation this December 2013. The other one, Olga Balkanova, studies some particular exponential sums called the GL(3) Kloosterman sums, which are directly linked to the Fourier coefficients of GL(3) automorphic forms via the trace formula. Her work will lead to several publications.

Professor Emmanuel Kowalski

ETH Zürich

Departement Mathematik

Rämistrasse 101

8092 Zürich Switzerland

kowalski@math.ethz.ch

http://www.math.ethz.ch/~kowalski/

Doctor Guillaume Ricotta

Institut de Mathématiques de Bordeaux

Université Bordeaux 1

351 cours de la Libération

33405 Talence Cedex France

Guillaume.Ricotta@math.u-bordeaux1.fr

http://www.math.u-bordeaux1.fr/~gricotta/