Modular forms and elliptic curves over the cubic field of discriminant -23

TL;DR

This paper computationally investigates the modularity of elliptic curves over the cubic field of discriminant -23 by explicitly computing cohomology of congruence subgroups of GL(2,O).

Contribution

It provides the first explicit cohomology computations for congruence subgroups over this specific cubic field, advancing understanding of elliptic curve modularity in this context.

Findings

Evidence supporting modularity of certain elliptic curves over the field

Explicit cohomology data for congruence subgroups of GL(2,O)

New computational techniques for arithmetic in cubic fields

Abstract

Let F be the cubic field of discriminant -23 and let O be its ring of integers. By explicitly computing cohomology of congruence subgroups of GL(2,O), we computationally investigate modularity of elliptic curves over F.