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Low-lying zeros of elliptic curve L-functions: Beyond the Ratios Conjecture

Published online by Cambridge University Press:  08 January 2016

DANIEL FIORILLI
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Ave, Ottawa, Ontario, K1N 6N5, Canada. e-mail: Daniel.Fiorilli@uottawa.ca
JAMES PARKS
Affiliation:
Department of Mathematics and Statistics, Concordia University, Montreal, QC, H3G 1M8, Canada. e-mail: james.parks@uleth.ca
ANDERS SÖDERGREN
Affiliation:
School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, U.S.A. e-mail: sodergren@math.ku.dk

Abstract

We study the low-lying zeros of L-functions attached to quadratic twists of a given elliptic curve E defined over $\mathbb{Q}$. We are primarily interested in the family of all twists coprime to the conductor of E and compute a very precise expression for the corresponding 1-level density. In particular, for test functions whose Fourier transforms have sufficiently restricted support, we are able to compute the 1-level density up to an error term that is significantly sharper than the square-root error term predicted by the L-functions Ratios Conjecture.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2016 

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