Fields of definition of elliptic curves with prescribed torsion

TL;DR

This paper proves that certain elliptic curves over quadratic and cubic fields with specific torsion subgroups are actually base changes of elliptic curves defined over the rationals, using modular curve analysis.

Contribution

It provides a modular curve-based proof that elliptic curves with specific torsion over quadratic and cubic fields are base changes from .

Findings

Elliptic curves over quadratic fields with C_{16} torsion are base changes from .

Elliptic curves over cubic fields with C_2 times C_{14} torsion are base changes from .

The study of modular curves and maps yields these classification results.

Abstract

We prove that all elliptic curves over quadratic fields with a subgroup isomorphic to $C_{16}$, as well as all elliptic curves over cubic fields with a subgroup isomorphic to $C_2\times C_{14}$, are base changes of elliptic curves defined over $\Q$. We obtain these results by studying geometric properties of modular curves and maps between modular curves, and then obtaining a modular description of these curves and maps.