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 k ≥ 2, level N with integer Fourier coefficients. In particular, we show that for such a form f and for any real number ǫ > 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−ǫ 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 ≥ 2.