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Modular elliptic curves over the field of twelfth roots of unity

Part of: Arithmetic algebraic geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 April 2016

Andrew Jones
Andrew Jones*
Affiliation:
School of Mathematics and Statistics, University of Sheffield, Hounsfield Road, SheffieldS3 7RH, United Kingdom email [email protected]

Abstract

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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.

MSC classification

Primary: 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11G05: Elliptic curves over global fields 11F80: Galois representations
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 19 , Issue 1 , 2016 , pp. 155 - 174
Copyright
© The Author 2016 

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