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On a non-Archimedean analogue of a question of Atkin and Serre

Yuri F Bilu, Sanoli Gun, Sunil L Naik

Abstract

In this article, we investigate a non-Archimedean analogue of a question of Atkin and Serre. More precisely, we derive lower bounds for the largest prime factor of non-zero Fourier coefficients of non-CM normalized cuspidal Hecke eigenforms of even weight $k \geq 2$, level $N$ with integer Fourier coefficients. In particular, we show that for such a form $f$ and for any real number $ε>0$, the largest prime factor of the $p$-th Fourier coefficient $a_f(p)$ of $f$, denoted by $P(a_f(p))$, satisfies $$ P(a_f(p)) ~>~ (\log p)^{1/8}(\log\log p)^{3/8 -ε} $$ for almost all primes $p$. This improves on earlier bounds. We also investigate a number field analogue of a recent result of Bennett, Gherga, Patel and Siksek about the largest prime factor of $a_f(p^m)$ for $m \geq 2$.

On a non-Archimedean analogue of a question of Atkin and Serre

Abstract

In this article, we investigate a non-Archimedean analogue of a question of Atkin and Serre. More precisely, we derive lower bounds for the largest prime factor of non-zero Fourier coefficients of non-CM normalized cuspidal Hecke eigenforms of even weight , level with integer Fourier coefficients. In particular, we show that for such a form and for any real number , the largest prime factor of the -th Fourier coefficient of , denoted by , satisfies for almost all primes . This improves on earlier bounds. We also investigate a number field analogue of a recent result of Bennett, Gherga, Patel and Siksek about the largest prime factor of for .
Paper Structure (19 sections, 21 theorems, 158 equations)

This paper contains 19 sections, 21 theorems, 158 equations.

Table of Contents

  1. Introduction and Statements of Results
  2. Preliminaries
  3. Applications of $\ell$-adic Galois representations and Chebotarev density theorem
  4. The Sato-Tate conjecture
  5. Height of an algebraic number
  6. Prerequisites from cyclotomic polynomials and other required results
  7. Largest prime factor of Fourier coefficients at primes
  8. Proof of Theorem \ref{['thm1']}
  9. Proof of Theorem \ref{['thm3']}
  10. Proof of Theorem \ref{['thm2']}
  11. Largest prime factor of $a_f(n)$
  12. Proof of Corollary \ref{['cafn']}
  13. Proof of Corollary \ref{['cafnG']}
  14. Proof of Corollary \ref{['cafnG2']}
  15. Largest prime factor of Fourier coefficients at prime powers
  16. ...and 4 more sections

Key Result

Theorem 1

Let $f$ be a non-CM normalized cuspidal Hecke eigenform of even weight $k \geq 2$ for $\Gamma_0(N)$ having integer Fourier coefficients $\{a_f(n) : n \in \mathbb{N}\}$ and let $\epsilon>0$ be a real number. Then we have for almost all primes $p$.

Theorems & Definitions (28)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Remark 1.1
  • Theorem 4
  • Corollary 5
  • Corollary 6
  • Corollary 7
  • Corollary 8
  • Theorem 9
  • ...and 18 more