When Kloosterman sums meet Hecke eigenvalues

• 成果类型: Article
• 署名作者: Xi, Ping
• 署名单位: Xi'an Jiaotong University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00924-y
• 发表日期: 2020
• 页码: 61-127
• 关键词: sato-tate conjecture; automorphic-forms; sieve

摘要

By elaborating a two-dimensional Selberg sieve with asymptotics and equidistributions of Kloosterman sums from l\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document}-adic cohomology, as well as a Bombieri-Vinogradov type mean value theorem for Kloosterman sums in arithmetic progressions, it is proved that for any given primitive Hecke-Maass cusp form of trivial nebentypus, the eigenvalue of the n-th Hecke operator does not coincide with the Kloosterman sum Kl(1,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {Kl}(1,n)$$\end{document} for infinitely many squarefree n with at most 100 prime factors. This provides a partial negative answer to a problem of Katz on modular structures of Kloosterman sums.