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

In his paper \cite{Atkinc}, Atkin pioneered computer investigations of divisibility properties of Fourier coefficients of the modular invariant by powers of $13,17,19$, and $23$. On the basis of these computations he formulated certain conjectures in \cite{Atkinc,AtkinH}. In particular, the question why similar congruence properties occur for these primes is posed in \cite{Atkinc}. We show how a combination of Serre’s theory of $p$-adic modular forms and Hida’s Control Theorem explains the phenomenon.

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Published 1 January 2006