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

Institut for Matematiske Fag

2 Citationer (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.

Originalsprog	Engelsk
Tidsskrift	Mathematika
Vol/bind	62
Udgave nummer	2
Sider (fra-til)	321- 336
ISSN	0025-5793
DOI	https://doi.org/10.1112/S0025579315000212
Status	Udgivet - 22 jan. 2016

Adgang til dokumentet

10.1112/S0025579315000212

Citationsformater

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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.",

author = "Imin Chen and Ian Kiming",

year = "2016",

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doi = "10.1112/S0025579315000212",

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journal = "Mathematika",

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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

Y1 - 2016/1/22

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.

U2 - 10.1112/S0025579315000212

DO - 10.1112/S0025579315000212

M3 - Journal article

SN - 0025-5793

VL - 62

SP - 321

EP - 336

JO - Mathematika

JF - Mathematika

IS - 2

ER -