Low-lying zeros of families of elliptic curves
Number Theory
2020-08-17 v3
Abstract
We study the low-lying zeros of various interesting families of elliptic curve L-functions. One application is an upper bound on the average analytic rank of the family of all elliptic curves. The upper bound obtained is less than two, which implies that a positive proportion of elliptic curves over the rationals have algebraic rank equal to analytic rank and finite Tate-Shafarevich group. These results are conditional on the Generalized Riemann Hypothesis.
Keywords
Cite
@article{arxiv.math/0406330,
title = {Low-lying zeros of families of elliptic curves},
author = {Matthew P. Young},
journal= {arXiv preprint arXiv:math/0406330},
year = {2020}
}
Comments
v2: Enhanced exposition, 56 pages. v3: One reference added and one sentence changed in the paragraph following Corollary 3.4