Infinite Families Of Isogeny-Torsion Graphs

TL;DR

This paper classifies isogeny-torsion graphs of elliptic curves over rationals, identifying which graphs correspond to infinitely many j-invariants, thus advancing understanding of elliptic curve isogeny structures.

Contribution

It provides a complete classification of isogeny-torsion graphs associated with elliptic curves over b4, determining which graphs occur infinitely often.

Findings

Identifies isogeny-torsion graphs with infinitely many j-invariants.

Classifies all possible isogeny-torsion graph structures over b4.

Establishes criteria linking graph structure to infinite families.

Abstract

Let $\mathcal{E}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined over $\mathbb{Q}$. The isogeny graph associated to $\mathcal{E}$ is a graph which has a vertex for each element of $\mathcal{E}$ and an edge for each $\mathbb{Q}$-isogeny of prime degree that maps one element of $\mathcal{E}$ to another element of $\mathcal{E}$, with the degree recorded as a label of the edge. The isogeny-torsion graph associated to $\mathcal{E}$ is the isogeny graph associated to $\mathcal{E}$ where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over $\mathbb{Q}$ of the corresponding elliptic curve. The main result of the article is a determination of which isogeny-torsion graphs associated to $\mathbb{Q}$-isogeny classes of elliptic curves defined over $\mathbb{Q}$ correspond to infinitely many $\textit{j}$-invariants.