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Title: On the low-lying zeros of Hasse–Weil L-functions for elliptic curves
Authors: Baier, Stephan
Zhao, Liangyi
Keywords: DRNTU::Science::Mathematics::Number theory
Issue Date: 2008
Source: 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.
Series/Report no.: Advances in Mathematics.
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-Shafarevic group. Statements of this flavor were known previously under the additional assumptions of GRH for Dirichlet L-functions and symmetric square L-functions which are removed in the present paper.
ISSN: 0001-8708
Schools: School of Physical and Mathematical Sciences 
Fulltext Permission: open
Fulltext Availability: With Fulltext
Appears in Collections:SPMS Journal Articles

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