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ON THE THETA OPERATOR FOR MODULAR FORMS MODULO PRIME POWERS

  • Imin Chen (a1) and Ian Kiming (a2)

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.

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