Moments of the error term in the Sato-Tate law for elliptic curves

JOURNAL OF NUMBER THEORY

卷:194

Article

Baier, Stephan¹  Prabhu, Neha² 

[1] Ramakrishna Mission Vivekananda Univ, Dept Math, PO Belur Math, Haora 711202, W Bengal, India

[2] Queens Univ, Kingston, ON K7K 3N6, Canada

关键词: Sato-Tate conjecture;    Elliptic curves;    Moments;    Central Limit Theorem;    Traces of Hecke operators;   

DOI  :  10.1016/j.jnt.2018.07.005

来源: Elsevier

【 摘 要 】

We derive new bounds for moments of the error in the Sato-Tate law over families of elliptic curves. Our estimates are stronger than those obtained in [4] and [5] for the first and second moment, but this comes at the cost of larger ranges of averaging. As applications, we deduce new almost-all results for the said errors and a conditional Central Limit Theorem on the distribution of these errors. Our method is different from those used in the above-mentioned papers and builds on recent work by the second-named author and K. Sinha [21] who derived a Central Limit Theorem on the distribution of the errors in the Sato-Tate law for families of cusp forms for the full modular group. In addition, identities by Birch and Melzak play a crucial rule in this paper. Birch's identities connect moments of coefficients of Hasse-Weil L-functions for elliptic curves with the Kronecker class number and further with traces of Hecke operators. Melzak's identity is combinatorial in nature. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free