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Moments and distribution of central \(L\)-values of quadratic twists of elliptic curves

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Abstract

We show that if one can compute a little more than a particular moment for some family of L-functions, then one has upper bounds of the conjectured order of magnitude for all smaller (positive, real) moments and a one-sided central limit theorem holds. We illustrate our method for the family of quadratic twists of an elliptic curve, obtaining sharp upper bounds for all moments below the first. We also establish a one sided central limit theorem supporting a conjecture of Keating and Snaith. Our work leads to a conjecture on the distribution of the order of the Tate-Shafarevich group for rank zero quadratic twists of an elliptic curve, and establishes the upper bound part of this conjecture (assuming the Birch-Swinnerton-Dyer conjecture).

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Acknowledgments

We are grateful to Brian Conrad for some helpful discussions, and to the referee for a careful reading of the paper.

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Correspondence to Maksym Radziwiłł.

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Dedicated to Peter Sarnak on the occasion of his sixty first birthday

The first author was partially supported by NSF grant DMS-1128155. The second author is partially supported by NSF grant DMS-1001068, and a Simons Investigator award from the Simons Foundation.

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Radziwiłł, M., Soundararajan, K. Moments and distribution of central \(L\)-values of quadratic twists of elliptic curves. Invent. math. 202, 1029–1068 (2015). https://doi.org/10.1007/s00222-015-0582-z

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  • DOI: https://doi.org/10.1007/s00222-015-0582-z

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