On the low-lying zeros of Hasse-Weil L-functions for elliptic curves

Citations of this article
Mendeley users who have this article in their library.
Get full text


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-Shafarevich group. Statements of this flavor were known previously [M.P. Young, Low-lying zeros of families of elliptic curves, J. Amer. Math. Soc. 19 (1) (2005) 205-250] under the additional assumptions of GRH for Dirichlet L-functions and symmetric square L-functions which are removed in the present paper. © 2008 Elsevier Inc. All rights reserved.




Baier, S., & Zhao, L. (2008). On the low-lying zeros of Hasse-Weil L-functions for elliptic curves. Advances in Mathematics, 219(3), 952–985. https://doi.org/10.1016/j.aim.2008.06.006

Register to see more suggestions

Mendeley helps you to discover research relevant for your work.

Already have an account?

Save time finding and organizing research with Mendeley

Sign up for free