The Average Analytic Rank of Elliptic Curves

Abstract

All the results in this paper are conditional on the Riemann hypothesis for the $L$-functions of elliptic curves. Under this assumption, we show that the average analytic rank of all elliptic curves over $\mathbb{Q}$ is at most 2, thereby improving a result of Brumer [2]. We also show that the average within any family of quadratic twists is at most $3/2$, improving a result of Goldfeld [3]. A third result concerns the density of curves with analytic rank at least $R$ and shows that the proportion of such curves decreases faster than exponentially as $R$ grows. The proofs depend on an analogue of Weil's ``explicit formula.''