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On Atkin and Swinnerton-Dyer congruences for noncongruence modular forms

Author: Jonas Kibelbek
Journal: Proc. Amer. Math. Soc. 142 (2014), 4029-4038
MSC (2010): Primary 11F33; Secondary 11S31, 15A03, 11B37
Published electronically: July 28, 2014
MathSciNet review: 3266975
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Abstract: In 1985, Scholl showed that Fourier coefficients of noncongruence cusp forms satisfy an infinite family of congruences modulo powers of $p$, providing a framework for understanding the Atkin and Swinnerton-Dyer congruences. We show that solutions to the weight-$k$ Scholl congruences can be rewritten, modulo the appropriate powers of $p$, as $p$-adic solutions of the corresponding linear recurrence relation. Finally, we show that there are spaces of cusp forms that do not admit any basis satisfying 3-term Atkin and Swinnerton-Dyer type congruences at supersingular places, settling a question raised by Atkin and Swinnerton-Dyer.

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Additional Information

Jonas Kibelbek
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
Address at time of publication: 3505 Sharonwood Road, Apt. 2D, Laurel, Maryland 20724

Keywords: Atkin and Swinnerton-Dyer congruences, $p$-adic congruences, noncongruence modular forms
Received by editor(s): February 2, 2012
Received by editor(s) in revised form: December 28, 2012
Published electronically: July 28, 2014
Additional Notes: This research was supported in part by NSF grant DMS-0801096 and NSA grant H98230-10-1-0195. Part of the research was done when the author was visiting the National Center for Theoretical Sciences in Hsinchu, Taiwan, and he thanks the NCTS for its hospitality. The author would like to thank Dr. Li for her encouragement and helpful suggestions, and is grateful for the referee’s helpful comments, suggestions, and patience.
Communicated by: Kathrin Bringmann
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The copyright for this article reverts to public domain 28 years after publication.