Low-lying zeros of families of elliptic curves

Abstract

We study the low-lying zeros of various interesting families of elliptic curve L-functions. One application is an upper bound on the average analytic rank of the family of all elliptic curves. The upper bound obtained is less than two, which implies that a positive proportion of elliptic curves over the rationals have algebraic rank equal to analytic rank and finite Tate-Shafarevich group. These results are conditional on the Generalized Riemann Hypothesis.