Matthew P. Young
Published 2004-06-16, updated 2005-04-06Version 3
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.
Comments: v2: Enhanced exposition, 56 pages. v3: One reference added and one sentence changed in the paragraph following Corollary 3.4
Low-lying zeros of L-functions for Quaternion Algebras
Low-lying zeros of elliptic curve L-functions: Beyond the ratios conjecture
Low-lying zeros of quadratic Dirichlet $L$-functions: Lower order terms for extended support
![QR code for arXiv:math/0406330 [math.NT]](http://oss.arxitics.com/images/favicon.svg)