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ON THE THETA OPERATOR FOR MODULAR FORMS MODULO PRIME POWERS

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We consider the classical theta operator θ on modular forms modulo pm and level N prime to p, where p is a prime greater than three. Our main result is that θ mod pm will map forms of weight k to forms of weight k+2+2pm−1(p−1) and that this weight is optimal in certain cases when m is at least two. Thus, the natural expectation that θ mod pm should map to weight k+2+pm−1(p−1) is shown to be false. The primary motivation for this study is that application of the θ operator on eigenforms mod pm corresponds to twisting the attached Galois representations with the cyclotomic character. Our construction of the θ-operator mod pm gives an explicit weight bound on the twist of a modular mod pm Galois representation by the cyclotomic character.
OriginalsprogEngelsk
TidsskriftMathematika
Vol/bind62
Udgave nummer2
Sider (fra-til)321- 336
ISSN0025-5793
DOI
StatusUdgivet - 2016

ID: 154005878