2-adic Galois images of non-CM isogeny-torsion graphs

TL;DR

This paper classifies the possible 2-adic Galois images associated with elliptic curves over  with no complex multiplication, based on their isogeny-torsion graphs, providing a comprehensive understanding of their Galois representations.

Contribution

It provides a complete classification of 2-adic Galois images for elliptic curves over  without CM within isogeny-torsion graphs, a novel comprehensive analysis.

Findings

Classified all 2-adic Galois images for non-CM elliptic curves over .

Connected Galois image types to isogeny-torsion graph structures.

Enhanced understanding of Galois representations in the context of elliptic curve isogenies.

Abstract

Let $\mathcal{E}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined over $\mathbb{Q}$ without CM. The isogeny graph associated to $\mathcal{E}$ is a graph which has a vertex for each elliptic curve in $\mathcal{E}$ and an edge for each $\mathbb{Q}$-isogeny of prime degree that maps one elliptic curve in $\mathcal{E}$ to another elliptic curve in $\mathcal{E}$, with the degree recorded as a label of the edge. An isogeny-torsion graph is an isogeny graph where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over $\mathbb{Q}$ of the corresponding elliptic curve. Then, the main statement of the article is a classification of the $2$-adic image of Galois that occurs at each vertex of all isogeny-torsion graphs consisting of elliptic curves defined over $\mathbb{Q}$ without CM.