Central values of L-functions encode essential arithmetic information in contexts ranging from distribution of primes to elliptic curves and arithmetic manifolds. In this talk, I will present asymptotic formulas for moments in families of twisted L-functions with all primitive characters modulo q, with a power saving in q.
On the one hand, we use the full power of spectral theory of GL(2) automorphic forms to treat a possibly highly unbalanced shifted convolution problem; on the other hand, we combine analytic number theory and input from algebraic geometry (including the Riemann Hypothesis for curves over finite fields and independence of Kloosterman sheaves) to prove estimates on bilinear forms in Kloosterman sums in critical ranges. The emphasis in this talk will be on explaining how these various methods fit together as well as how algebraic geometry naturally enters the analytic problem of asymptotic evaluation of moments.
In addition to providing statistical and intrinsic information about the underlying family of automorphic forms, asymptotics of moments are an essential ingredient in analytic approaches to questions of arithmetic importance such as upper bounds, nonvanishing, or extreme values, and I will also survey several of our applications. This is joint work with Blomer, Fouvry, Kowalski, Michel, and Sawin.