On Atkin and SwinnertonDyer congruences for noncongruence modular forms
Author:
Jonas Kibelbek
Journal:
Proc. Amer. Math. Soc.
MSC (2010):
Primary 11F33; Secondary 11S31, 15A03, 11B37
Published electronically:
July 28, 2014
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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 , providing a framework for understanding the Atkin and SwinnertonDyer congruences. We show that solutions to the weight Scholl congruences can be rewritten, modulo the appropriate powers of , as 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 3term Atkin and SwinnertonDyer type congruences at supersingular places, settling a question raised by Atkin and SwinnertonDyer.
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 E. J. Ditters, Hilbert functions and Witt functions. An identity for congruences of Atkin and of SwinnertonDyer type, Math. Z. 205 (1990), no. 2, 247278. MR 1076132 (92b:14025), http://dx.doi.org/10.1007/BF02571239
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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
Email:
jckibelbek@gmail.com
DOI:
http://dx.doi.org/10.1090/S000299392014121629
PII:
S 00029939(2014)121629
Keywords:
Atkin and SwinnertonDyer 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 DMS0801096 and NSA grant H982301010195. 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
Article copyright:
© Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
