ON THE THETA OPERATOR FOR MODULAR FORMS MODULO PRIME POWERS

Imin Chen; Ian Kiming

doi:10.1112/S0025579315000212

ON THE THETA OPERATOR FOR MODULAR FORMS MODULO PRIME POWERS

Imin Chen, Ian Kiming

Department of Mathematical Sciences

2 Citations (Scopus)

Abstract

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.

Original language	English
Journal	Mathematika
Volume	62
Issue number	2
Pages (from-to)	321- 336
ISSN	0025-5793
DOIs	https://doi.org/10.1112/S0025579315000212
Publication status	Published - 22 Jan 2016

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title = "ON THE THETA OPERATOR FOR MODULAR FORMS MODULO PRIME POWERS",

abstract = "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.",

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T1 - ON THE THETA OPERATOR FOR MODULAR FORMS MODULO PRIME POWERS

AU - Chen, Imin

AU - Kiming, Ian

PY - 2016/1/22

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N2 - 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.

AB - 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.

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DO - 10.1112/S0025579315000212

M3 - Journal article

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JO - Mathematika

JF - Mathematika

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