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

Abstract

In 1916 Ramanujan observed a remarkable congruence: τ ( n ) ≡ σ 11 ( n ) mod 691 \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 12 and the Eisenstein series of the same weight modulo the numerator of the Bernoulli number B 12 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.