Modular Elliptic Curves over the Field of Twelfth Roots of Unity

TL;DR

This paper investigates automorphic forms for GL(2) over a quartic CM field generated by twelfth roots of unity, computing cohomology and Hecke actions to identify new modular elliptic curves over this field.

Contribution

It provides the first known examples of modular elliptic curves over the field generated by twelfth roots of unity, using computational cohomology and Galois representation techniques.

Findings

First known modular elliptic curves over this field identified.

Computed automorphic forms and Hecke actions for the specific field.

Established a method for studying elliptic curves over complex CM fields.

Abstract

In this article we perform an extensive study of the spaces of automorphic forms for GL(2) of weight two and level N, for N an ideal in the ring of integers of the quartic CM field generated by the twelfth roots of unity. This study is conducted through the computation of the group cohomology of the congruence subgroup of upper triangular matrices modulo N, and the corresponding Hecke action which this space admits. 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.