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

Baier, Stephan and Zhao, Liangyi (2008) On the low-lying zeros of Hasse-Weil L-functions for elliptic curves. Advances in Mathematics, 219 (3). pp. 952-985. ISSN 0001-8708

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Abstract

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.

Item Type:	Article
Faculty \ School:	Faculty of Science > School of Mathematics (former - to 2024)
UEA Research Groups:	Faculty of Science > Research Groups > Number Theory, Ergodic Theory and Dynamical Systems (former - to 2013) Faculty of Science > Research Groups > Number Theory (former - to 2017)
Depositing User:	Pure Connector
Date Deposited:	07 Sep 2013 05:13
Last Modified:	28 Mar 2025 05:15
URI:	https://ueaeprints.uea.ac.uk/id/eprint/43207
DOI:	10.1016/j.aim.2008.06.006

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