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

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 $k\ge 2$, level $N$ with integer Fourier coefficients. In particular, we show that for such a form $f$ and for any real number $\epsilon >0$, the largest prime factor of the $p$-th Fourier coefficient $a_f(p)$ of $f$, denoted by $P(a_f(p))$, satisfies $$P(a_f(p))>(\log p{)}^{1\mathrm{/}8}(\log \log p{)}^{3\mathrm{/}8-\epsilon }$$ 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 $m\ge 2$.