Overview

The workshop will explore statistical aspects of various questions in number theory. One theme concerns statistics of zeros of L-functions, which since the work of Montgomery, are known to display striking similarities to the statistics of eigenvalues of random matrices. It is predicted by the Katz and Sarnak philosophy that in a suitable limit, the statistics for the zeroes in many families of L-functions follow distribution laws of random matrices. This was proven by Katz and Sarnak for several families of curves of fixed genus over finite fields, in the limit of large finite fields. Other statistics can be obtained by considering families of L-functions over a fixed finite field, when the genus gets large. This approach led to some striking recent work where the results for L-functions of curves over finite fields can be used to prove results for L-functions over number fields. Another theme of the workshop will be statistics obtained by fixing a global object over the rationals and looking at properties of the reductions modulo primes p. One example is the Sato-Tate conjecture, concerning the distribution of the reduced normalised trace a_p(E)/2√p of a fixed elliptic curve E over Q. This can also be rephrased in the language of random matrix theory, the Sato-Tate measure being the Haar measure on the group of 2-by-2 symplectic matrices.

The goal of the workshop is to explore refinements and recent breakthroughs on these subjects.

The workshop will also include a lecture series by the Andre Aisenstadt chaire Zeév Rudnick.