A conditional determination of the average rank of elliptic curves

作者：Fiorilli Daniel^*

来源：Journal of the London Mathematical Society-Second Series, 2016, 94(3): 767-792.

DOI：10.1112/jlms/jdw058

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 12 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.

• 出版日期2016-12