Prime divisors of non-zero Fourier coefficients of Hecke eigenforms [HBNI Th 227]

Abstract

The focal point of our thesis is to study prime divisors of non-zero Fourier coefficients of Hecke eigenforms. Let f be a non-CM normalized Hecke eigenform and for any natural number n, let a f (n) be its n-th Fourier coefficient. It is well known that a f (n)’s are algebraic integers and the field K f generated by all its Fourier coefficients is a number field. To begin, we establish a lower bound for the number of distinct prime factors and radicals of a f (n) for infinitely many natural numbers n. Divisibility properties of Lucas sequences play a major role in proving this.