Mathematical Research Letters

Volume 13 (2006)

Number 4

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

Pages: 549 – 555



P. Guerzhoy (Temple University)


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.

Published 1 January 2006