2-Adic Galois Images of Non-CM Isogeny-Torsion Graphs Defined over ℚ

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 Galois image that occurs at each vertex of all isogeny-torsion graphs consisting of elliptic curves defined over $\mathbb{Q}$ without CM.

KEYWORDS:

• elliptic curves
• Galois representations
• rational points
• isogeny graphs

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 11F80
• Secondary: 11G05
• 11G15
• 14H52

Acknowledgments

The author would like to express his utmost gratitude to Álvaro Lozano-Robledo, Jeremy Rouse, Drew Sutherland, and David Zureick-Brown for their patience and many helpful conversations on this topic. The author would also like to thank the referee for many comments that improved the paper.