On the low-lying zeros of Hasse-Weil L-functions for Elliptic Curves
by
Liangyi Zhao
Nanyang Technological University
Coauthors: Stephan Baier

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 (strictly less than 2) for the average analytic rank of the elliptic curves in the family under consideration. This upper bound 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 established previously by M. P. Young under the additional assumptions of GRH for Dirichlet L-functions and symmetric square L-functions which are removed here.

Date received: April 15, 2008