Hecke operators for weakly holomorphic modular forms and supersingular congruences

Author:
P. Guerzhoy

Journal:
Proc. Amer. Math. Soc. **136** (2008), 3051-3059

MSC (2000):
Primary 11F37, 11F33

DOI:
https://doi.org/10.1090/S0002-9939-08-09277-0

Published electronically:
April 29, 2008

MathSciNet review:
2407067

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the action of Hecke operators on weakly holomorphic modular forms and a Hecke-equivariant duality between the spaces of holomorphic and weakly holomorphic cusp forms. As an application, we obtain congruences modulo supersingular primes, which connect Hecke eigenvalues and certain singular moduli.

**1.**Asai, T.; Kaneko, M.; Ninomiya, H., Zeros of certain modular functions and an application, Comment. Math. Univ. St. Paul. 46 (1997), no. 1, 93-101. MR**1448475 (98e:11052)****2.**G. Bol, Invarianten linearer differentialgleichungen, Abh. Math. Sem. Univ. Hamburg 16 (1949), 1-28. MR**0033411 (11:437a)****3.**Bringmann, K.; Ono, K., Lifting elliptic cusp forms to Maass forms with an application to partitions, Proceedings of the National Academy of Sciences, USA, 104 (2007), no. 10, 3725-3731. MR**2301875****4.**Bringmann, K.; Ono, K., Dyson's ranks and Maass forms, Annals of Mathematics, to appear.**5.**Bruinier, Jan H.; Kohnen, Winfried; Ono, Ken, The arithmetic of the values of modular functions and the divisors of modular forms, Compos. Math. 140 (2004), no. 3, 552-566. MR**2041768 (2005h:11083)****6.**Hida, Haruzo, Elementary theory of -functions and Eisenstein series, London Mathematical Society Student Texts, 26, Cambridge University Press, Cambridge, 1993. MR**1216135 (94j:11044)****7.**Kohnen, W.; Zagier, D., Modular forms with rational periods, Modular forms (Durham, 1983), 197-249, Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984. MR**803368 (87h:11043)****8.**Lang, Serge, Introduction to modular forms. With appendices by D. Zagier and Walter Feit. Corrected reprint of the 1976 original. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 222, Springer-Verlag, Berlin, 1995. MR**1363488 (96g:11037)****9.**Ono, Ken, The web of modularity: Arithmetic of the coefficients of modular forms and -series. CBMS Regional Conference Series in Mathematics, 102. Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 2004. MR**2020489 (2005c:11053)****10.**Zagier, D., Traces of singular moduli, motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), 211-244, Int. Press Lect. Ser., 3, I, Int. Press, Somerville, MA, 2002. MR**1977587 (2004h:11037)**

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Additional Information

**P. Guerzhoy**

Affiliation:
Department of Mathematics, University of Hawaii at Manoa, 2565 McCarthy Mall, Honolulu, Hawaii 96822-2273

Email:
pavel@math.hawaii.edu

DOI:
https://doi.org/10.1090/S0002-9939-08-09277-0

Received by editor(s):
April 23, 2007

Received by editor(s) in revised form:
July 16, 2007

Published electronically:
April 29, 2008

Additional Notes:
This work is supported by NSF grant DMS-0700933

Communicated by:
Ken Ono

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.