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

TL;DR

This paper establishes an unconditional density theorem for low-lying zeros of Hasse-Weil L-functions of elliptic curves, leading to new bounds on average ranks and properties of elliptic curves without relying on GRH.

Contribution

It provides the first unconditional density theorem for these zeros and derives bounds on average ranks without assuming GRH for related L-functions.

Findings

Unconditional density theorem for low-lying zeros

Majorant of 27/14 for average rank of elliptic curves

Positive proportion of elliptic curves with algebraic and analytic ranks equal

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.