On a non-Archimedean analogue of a question of Atkin and Serre
type: Item
instance of: scholarly article
title: On a non-Archimedean analogue of a question of Atkin and Serre
author: Yuri Bilu
author: Sanoli Gun
author: Sunil L. Naik
published in: Mathematische Annalen
publication date: 2024-05-17T00:00:00Z
review text: Let \(f\) be a non-CM normalized cuspidal Hecke eigenform of even weight \(k\geq 2\) for \(\Gamma_0(N)\) having integer Fourier coefficients \(a_f(n)\) (for \(n=1,2,\ldots\)). Let \(\varepsilon > 0\) be a real number. The authors prove (Theorem 1) the following inequality\N\[\NP(a_f(p)) > (\log p)^{1/8} (\log\log p)^{3/8 - \varepsilon}\N\]\Nfor almost all primes \(p\), where \(P(r)\) denote the largest prime factor of a positive integer \(r\). This improves on earlier bounds. Conditionally on GRH, they prove a stronger result (Theorem 3).\N\NThey also investigate a number field analogue (Theorem 9) of a recent result of \textit{M. A. Bennett} et al. [Math. Ann. 382, No. 1--2, 203--238 (2022; Zbl 1525.11034)] about the largest prime factor of \(a_f(p^m)\) for \(m\geq 2\).
reviewed by: Andrzej Dąbrowski
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11N56
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 11N36
Mathematics Subject Classification ID: 11R45
zbMATH DE Number: 7850201
zbMATH Keywords: question of Atkin and Serre
zbMATH Keywords: cuspidal Hecke eigenform
zbMATH Keywords: Galois representation
zbMATH Keywords: Chebotarev density theorem
zbMATH Keywords: Sato-Tate conjecture
MaRDI profile type: Publication
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DOI: 10.1007/S00208-023-02686-8
description: scientific article; zbMATH DE number 7850201