Central limit theorems for elliptic curves and modular forms with smooth weight functions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:485

Article

Baier, Stephan¹  Prabhu, Neha²  Sinha, Kaneenika³ 

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

[2] Inst Math Sci, CIT Campus, Chennai 600113, Tamil Nadu, India

[3] USER Pune, Dr Homi Bhabha Rd, Pune 411008, Maharashtra, India

关键词: Central Limit Theorems;    Modular forms;    Elliptic curves;    Sato-Tate law;   

DOI  :  10.1016/j.jmaa.2019.123709

来源: Elsevier

【 摘 要 】

In [11], the second and third-named authors established a Central Limit Theorem for the error term in the Sato-Tate law for families of modular forms. This method was adapted to families of elliptic curves in [3] by the first and second-named authors. In this context, a Central Limit Theorem was established only under a strong hypothesis going beyond the Riemann Hypothesis. In the present paper, we consider a smoothed version of the Sato-Tate conjecture, which allows us to overcome several limitations. In particular, for the smoothed version, we are able to establish a Central Limit Theorem for much smaller families of modular forms, and we succeed in proving a theorem of this type for families of elliptic curves under the Riemann Hypothesis for L-functions associated to Hecke eigenforms for the full modular group. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free