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Modular elliptic curves over the field of twelfth roots of unity

Published online by Cambridge University Press:  01 April 2016

Andrew Jones
Affiliation:
School of Mathematics and Statistics, University of Sheffield, Hounsfield Road, Sheffield S3 7RH, United Kingdom email andrew.j.jones@cantab.net
Corresponding
E-mail address:

Abstract

In this paper we perform an extensive study of the spaces of automorphic forms for $\text{GL}_{2}$ of weight $2$ and level $\mathfrak{n}$ , for $\mathfrak{n}$ an ideal in the ring of integers of the quartic CM field $\mathbb{Q}({\it\zeta}_{12})$ of twelfth roots of unity. This study is conducted through the computation of the Hecke module $H^{\ast }({\rm\Gamma}_{0}(\mathfrak{n}),\mathbb{C})$ , and the corresponding Hecke action. Combining this Hecke data with the Faltings–Serre method for proving equivalence of Galois representations, we are able to provide the first known examples of modular elliptic curves over this field.

Supplementary materials are available with this article.

Type
Research Article
Copyright
© The Author 2016 

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