Moments and distribution of central (formula presented. )-values of quadratic twists of elliptic curves

Maksym Radziwiłł*, K. Soundararajan

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

41 Scopus citations

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).

Original languageEnglish (US)
Pages (from-to)1029-1068
Number of pages40
JournalInventiones Mathematicae
Volume202
Issue number3
DOIs
StatePublished - Dec 1 2015

ASJC Scopus subject areas

  • General Mathematics

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