Inventiones mathematicae

, Volume 202, Issue 3, pp 1029–1068

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


DOI: 10.1007/s00222-015-0582-z

Cite this article as:
Radziwiłł, M. & Soundararajan, K. Invent. math. (2015) 202: 1029. doi:10.1007/s00222-015-0582-z


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).

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  2. 2.Department of MathematicsRutgers University - Hill CenterPiscatawayUSA
  3. 3.Department of MathematicsStanford UniversityStanfordUSA

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