On a non-Archimedean analogue of a question of Atkin and Serre
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866909104924000256 |
|---|---|
| author | Bilu, Yuri F Gun, Sanoli Naik, Sunil L |
| author_facet | Bilu, Yuri F Gun, Sanoli Naik, Sunil L |
| contents | 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 $m \geq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07943 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a non-Archimedean analogue of a question of Atkin and Serre Bilu, Yuri F Gun, Sanoli Naik, Sunil L Number Theory 11F11, 11F30, 11F80, 11N56, 11N05, 11N36, 11R45 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 $m \geq 2$. |
| title | On a non-Archimedean analogue of a question of Atkin and Serre |
| topic | Number Theory 11F11, 11F30, 11F80, 11N56, 11N05, 11N36, 11R45 |
| url | https://arxiv.org/abs/2402.07943 |