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.
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机译：我们考虑模数形式为p（m）的经典theta算子theta和p的第N级素数，其中p是大于3的素数。我们的主要结果是，theta mod pm将权重k的形式映射为权重k + 2 + 2p（m）1（p-1）的形式，并且该权重在m至少为2的某些情况下是最佳的。因此，θmod pm应该映射到权重k + 2 + p（m-1）（p-1）的自然期望被证明是错误的。这项研究的主要动机是，theta算子在特征形式mod pm上的应用对应于扭曲附加的Galois表示形式并具有环原子特征。我们的theta-operator mod pm的构造通过模态特征在模块化mod pm Galois表示的扭曲上给出了明确的权重约束。
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