On a conjecture of Atkin for the primes 13, 17, 19, and 23

Abstract

In his paper (2), Atkin pioneered computer investigations of divisibility prop- erties of Fourier coecien ts of the modular invariant by powers of 13; 17; 19, and 23. On the basis of these computations he formulated certain conjectures in (2, 3). In particular, the question why similar congruence properties occur for these primes is posed in (2). We show how a combination of Serre's theory of p-adic modular forms and Hida's Control Theorem explains the phenomenon.