ON THE THETA OPERATOR FOR MODULAR FORMS MODULO PRIME POWERS

We consider the classical theta operator theta on modular forms modulo p(m) and level N prime to p, where p is a prime greater than three. Our main result is that theta mod pm will map forms of weight k to forms of weight k + 2 + 2p(m) 1(p - 1) and that this weight is optimal in certain cases when m is at least two. Thus, the natural expectation that theta mod pm should map to weight k + 2 + p(m-1) (p - 1) is shown to be false. The primary motivation for this study is that application of the theta operator on eigenforms mod pm corresponds to twisting the attached Galois representations with the cyclotomic character. Our construction of the theta-operator mod pm gives an explicit weight bound on the twist of a modular mod pm Galois representation by the cyclotomic character.