On the distribution of analytic ranks of elliptic curves

Abstract

In this paper, under GRH for elliptic L-functions, we give an upper bound for the probability for an elliptic curve with analytic rank ≤ a for a ≥ 11, and also give an upper bound of n-th moments of analytic ranks of elliptic curves. These are applications of counting elliptic curves with local conditions, for example, having good reduction at p.