  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.
