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On Ramanujan congruences for modular forms of integral and half-integral weights

Author(s): B. Datskovsky; P. Guerzhoy
Journal: Proc. Amer. Math. Soc. 124 (1996), 2283-2291.
MSC (1991): Primary 11F33; Secondary 11F30, 11F32, 11F37
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Abstract: In 1916 Ramanujan observed a remarkable congruence: $\tau (n)\equiv \sigma _{11}(n) \quad \bmod \, 691$. The modern point of view is to interpret the Ramanujan congruence as a congruence between the Fourier coefficients of the unique normalized cusp form of weight $12$ and the Eisenstein series of the same weight modulo the numerator of the Bernoulli number $B_{12}$. In this paper we give a simple proof of the Ramanujan congruence and its generalizations to forms of higher integral and half-integral weights.

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Additional Information:

B. Datskovsky
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
Email: bdats@euclid.math.temple.edu

P. Guerzhoy
Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, 32000 Haifa, Israel
Address at time of publication: Fakultät für Mathematik und Informatik, Universität Mannheim, D-6800 Mannheim 1, Germany
Email: pasha@techunix.technion.ac.il, pasha@euklid.math.uni-mannheim.de

DOI: 10.1090/S0002-9939-96-03334-5
PII: S 0002-9939(96)03334-5
Received by editor(s): May 15, 1994
Additional Notes: The first author's research was supported by a Fulbright fellowship.
Communicated by: Dennis A. Hejhal
Copyright of article: Copyright 1996, American Mathematical Society

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