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arXiv · 2001.05616

A classification of isogeny-torsion graphs of $\mathbb{Q}$-isogeny classes of elliptic curves

Garen Chiloyan·Álvaro Lozano-Robledo
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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 elliptic curve in the $\mathbb{Q}$-isogeny class $\mathcal{E}$, and an edge for each cyclic $\mathbb{Q}$-isogeny of prime degree between elliptic curves in the isogeny class, with the degree recorded as a label of the edge. In this paper, we define an isogeny-torsion graph to be 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 result of the article is a classification of all the possible isogeny-torsion graphs that occur for $\mathbb{Q}$-isogeny classes of elliptic curves defined over the rationals.

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Garen Chiloyan, Álvaro Lozano-Robledo. 2020-01-16. A classification of isogeny-torsion graphs of $\mathbb{Q}$-isogeny classes of elliptic curves. https://doi.org/10.1112/tlm3.12024

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