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

• 成果类型: Article
• 署名作者: Radziwill, Maksym; Soundararajan, K.
• 署名单位: Institute for Advanced Study - USA; Stanford University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-015-0582-z
• 发表日期: 2015
• 页码: 1029-1068
• 关键词: random-matrix theory; Lower bounds; zeta

摘要

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).