Texas A&M Number Theory Seminar

Abstract: We will survey some results on random matrix theory and families of L-functions.  The Katz-Sarnak philosophy dictates that to any reasonable (e.g., geometrically defined) family of automorphic L-functions one can attach a classical compact Lie group (orthogonal, unitary or symplectic). Low-lying critical zero statistics correspond to eigenvalue statistics in this setting. Our emphasis in this talk is on the study of the effect of a natural representation-theoretical operation on families, namely the Rankin-Selberg convolution, on the underlying critical zero statistics. Supported by some previously known and some new examples of GL(2) families, we present a conjecture on the underlying symmetry of automorphic families for GL(n).  Our examples involve convolutions of symmetric-power families of L-functions attached to modular forms of varying weight, as well as L-functions of one-parameter families of elliptic curves. This is joint work with S.J. Miller, Brown University.