Families of L-Functions and Their Symmetry

Abstract

A few years ago the first-named author proposed a working definition of a family of automorphic L-functions. Then the work by the second and third-named authors on the Sato-Tate equidistribution for families made it possible to give a conjectural answer for the universality class introduced by Katz and the first-named author for the distribution of the zeros near s = 1/2. In this article we develop these ideas fully after introducing some structural invariants associated to the arithmetic statistics of a family.