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arXiv · 1405.5110

Low-lying zeros of elliptic curve L-functions: Beyond the ratios conjecture

Daniel Fiorilli·James Parks·Anders Södergren
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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.

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Daniel Fiorilli, James Parks, Anders Södergren. 2014-05-20. Low-lying zeros of elliptic curve L-functions: Beyond the ratios conjecture. https://doi.org/10.1017/s0305004115000730

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