Towards an 'average' version of the Birch and Swinnerton-Dyer Conjecture

TL;DR

This paper improves understanding of the distribution of zeros of L-functions associated with elliptic curves by averaging over families, providing bounds that support the Birch and Swinnerton-Dyer conjecture and related theories.

Contribution

It introduces a new averaging method over elliptic curve families to bound zeros near the central point at the expected scale.

Findings

Established non-trivial bounds for average zeros in small intervals

Provided evidence supporting the Birch and Swinnerton-Dyer conjecture

Connected results with predictions from random matrix theory

Abstract

The Birch and Swinnerton-Dyer conjecture states that the rank of the Mordell-Weil group of an elliptic curve E equals the order of vanishing at the central point of the associated L-function L(s,E). Previous investigations have focused on bounding how far we must go above the central point to be assured of finding a zero, bounding the rank of a fixed curve or on bounding the average rank in a family. Mestre showed the first zero occurs by O(1/loglog(N_E)), where N_E is the conductor of E, though we expect the correct scale to study the zeros near the central point is the significantly smaller 1/log(N_E). We significantly improve on Mestre's result by averaging over a one-parameter family of elliptic curves, obtaining non-trivial upper and lower bounds for the average number of normalized zeros in intervals on the order of 1/log(N_E) (which is the expected scale). Our results may be interpreted as providing further evidence in support of the Birch and Swinnerton-Dyer conjecture, as well as the Katz-Sarnak density conjecture from random matrix theory (as the number of zeros predicted by random matrix theory lies between our upper and lower bounds). These methods may be applied to additional families of L-functions.