A conditional determination of the average rank of elliptic curves

Under a hypothesis which is slightly stronger than the Riemann Hypothesis for elliptic curve $L$-functions, we show that both the average analytic rank and the average algebraic rank of elliptic curves in families of quadratic twists are exactly $\frac 12$. As a corollary we obtain that under this last hypothesis, the Birch and Swinnerton-Dyer Conjecture holds for almost all curves in our family, and that asymptotically one half of these curves have algebraic rank $0$, and the remaining half $1$. We also prove an analogous result in the family of all elliptic curves. A way to interpret our results is to say that nonreal zeros of elliptic curve $L$-functions in a family have a direct influence on the average rank in this family. Results of Katz-Sarnak and of Young constitute a major ingredient in the proofs.