Mathematical Research Letters

Volume 13 (2006)

Number 4

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

Pages: 549 – 555

DOI: http://dx.doi.org/10.4310/MRL.2006.v13.n4.a5

Author

P. Guerzhoy (Temple University)

Abstract

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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