On the Low-lying zeros of Hasse-Weil L-functions for Elliptic Curves - Mathematics > Number TheoryReport as inadecuate




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Abstract: In this paper, we obtain an unconditional density theorem concerning thelow-lying zeros of Hasse-Weil L-functions for a family of elliptic curves. Fromthis together with the Riemann hypothesis for these L-functions, we infer themajorant of 27-14 which is strictly less than 2 for the average rank of theelliptic curves in the family under consideration. This upper bound for theaverage rank enables us to deduce that, under the same assumption, a positiveproportion of elliptic curves have algebraic ranks equaling their analyticranks and finite Tate-Shafarevic group. Statements of this flavor were knownpreviously under the additional assumptions of GRH for Dirichlet L-functionsand symmetric square L-functions which are removed in the present paper.



Author: Stephan Baier, Liangyi Zhao

Source: https://arxiv.org/







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