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

Abstract

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.

arXiv comment23 pages. The majorant of the average analytic rank of all elliptic curves have been improved to 27/14 instead of the 241/122 in the previous version of the paper. To appear in Adv. Math

References in corpus (3)

• Low-lying zeros of families of elliptic curves
• Mean values with cubic characters
• Bombieri-Vinogradov Type Theorem for Sparse Sets of Moduli

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• One-level density and non-vanishing for cubic $L$-functions over the Eisenstein field
• Lower order terms of the one level density of a family of quadratic Hecke $L$-functions
• Low-lying zeros of a family of quadratic Hecke $L$-functions via ratios conjecture
• One level density of low-lying zeros of quadratic Hecke $L$-functions to prime moduli
• Extending the unconditional support in an Iwaniec-Luo-Sarnak family