On the distribution of analytic ranks of elliptic curves

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