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We consider the classical theta operator
on modular forms modulo
is a prime greater than three. Our main result is that
will map forms of weight
to forms of weight
and that this weight is optimal in certain cases when
is at least two. Thus, the natural expectation that
should map to weight
is shown to be false. The primary motivation for this study is that application of the
operator on eigenforms mod
corresponds to twisting the attached Galois representations with the cyclotomic character. Our construction of the
gives an explicit weight bound on the twist of a modular mod
Galois representation by the cyclotomic character.
Let p be a prime number. Let k be a field of characteristic different from p and containing the p-th roots of unity. Let be a finite group. Let L/k be a finite normal extension with Galois group and let c be a 2-cocycle on with coefficients in , where acts trivially on By Emb(L/k, c) we denote the question of the existence of a finite normal extension M of k, such that M contains L, such that [M: L] = p, and such that, denoting by the Galois group of M/k, the extension is given by the class of c.
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