On Galois Embedding Problems Arising from 3-Torsion of Elliptic Curves

TL;DR

This paper investigates Galois embedding problems related to the 3-torsion points of elliptic curves over rationals, extending the correspondence to all possible mod 3 Galois representation images.

Contribution

It extends the understanding of Galois embedding problems for elliptic curves by classifying all possible images of mod 3 Galois representations and relating solvability to elliptic curve families.

Findings

Solvability of embedding problems is linked to infinitely many elliptic curves with specific 3-division fields.

Classifies all possible images of mod 3 Galois representations for elliptic curves over $Q$.

Establishes equivalence between solvability and existence of certain elliptic curves in the cyclotomic case.

Abstract

We study Galois embedding problems arising from the 3-torsion of elliptic curves defined over $\mathbb{Q}$, extending the correspondence to all possible images of mod 3 Galois representations; namely, $\operatorname{GL}_2(\mathbb{F}_3),SD_{16},D_6,D_4$ and $C_2^2$. In the cyclotomic case, we show that solvability of these embedding problems is equivalent to the existence of infinitely many elliptic curves whose 3-division fields provide the corresponding solutions.