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Daniel Fiorilli, A conditional determination of the average rank of elliptic curves, Journal of the London Mathematical Society, Volume 94, Issue 3, December 2016, Pages 767–792, https://doi.org/10.1112/jlms/jdw058
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Abstract
Under a hypothesis that is stronger than the Riemann Hypothesis for elliptic curve |$L$|-functions, we show that both average analytic and algebraic ranks of elliptic curves in families of quadratic twists are exactly |$\frac {1}{2}$|. As a corollary we obtain that, under this last hypothesis, the Birch and Swinnerton-Dyer Conjecture holds for almost all curves, and that asymptotically one-half have algebraic rank 0, and the remaining half 1. We also prove an analogous result in the family of all elliptic curves. The proof uses results of Katz–Sarnak and Young on the 1-level density of zeros of elliptic curve |$L$|-functions. In essence, we show that, under a hypothesis analogous to Montgomery's Conjecture, a density result with limited support on low-lying zeros of |$L$|-functions is sufficient to determine the average rank.
