On the Low-lying zeros of Hasse-Weil L-functions for Elliptic Curves

Details On the Low-lying zeros of Hasse-Weil L-functions for Elliptic Curves On the Low-lying zeros of Hasse-Weil L-functions for Elliptic Curves

2007-08-22

arxiv.org/abs/0708.2987v4

Adv. Math., Vol. 219, No. 3, 2008, pp. 952-985.

Mathematics

Number Theory

23 pages. The majorant of the average analytic rank of all elliptic curves have been improved to 27/14 instead of the 241/122

Scientific paper

10.1016/j.aim.2008.06.006

In this paper, we obtain an unconditional density theorem concerning the low-lying zeros of Hasse-Weil L-functions for a family of elliptic curves. From this together with the Riemann hypothesis for these L-functions, we infer the majorant of 27/14 (which is strictly less than 2) for the average rank of the elliptic curves in the family under consideration. This upper bound for the average rank enables us to deduce that, under the same assumption, a positive proportion of elliptic curves have algebraic ranks equaling their analytic ranks and finite Tate-Shafarevic group. Statements of this flavor were known previously under the additional assumptions of GRH for Dirichlet L-functions and symmetric square L-functions which are removed in the present paper.

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