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

Datskovsky, B.; Guerzhoy, P.

doi:10.1090/S0002-9939-96-03334-5

On Ramanujan congruences for modular forms of integral and half-integral weights
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by B. Datskovsky and P. Guerzhoy PDF

Proc. Amer. Math. Soc. 124 (1996), 2283-2291 Request permission

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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