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# Texas A&M Number Theory Seminar

##
Department of Mathematics

Milner 317

Wednesdays, 12:30-1:30 PM

Eduardo
Duenez

University of Texas at San Antonio
**Wednesday, December 6**

Milner 317, 12:30 PM

**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.