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ON THE THETA OPERATOR FOR MODULAR FORMS MODULO PRIME POWERS

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  22 January 2016

Imin Chen  and
Ian Kiming
Imin Chen
Affiliation:
Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, B.C., V5A 1S6, Canada email [email protected]
Ian Kiming
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark email [email protected]
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Abstract

We consider the classical theta operator ${\it\theta}$ 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 ${\it\theta}$ mod $p^{m}$ 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 ${\it\theta}$ mod $p^{m}$ 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 ${\it\theta}$ operator on eigenforms mod $p^{m}$ corresponds to twisting the attached Galois representations with the cyclotomic character. Our construction of the ${\it\theta}$-operator mod $p^{m}$ gives an explicit weight bound on the twist of a modular mod $p^{m}$ Galois representation by the cyclotomic character.

MSC classification

Primary: 11F33: Congruences for modular and $p$-adic modular forms 11F80: Galois representations
Type
Research Article
Information
Mathematika , Volume 62 , Issue 2 , 2016 , pp. 321 - 336
Copyright
Copyright © University College London 2016 

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