STNB2023(36th edition)
Parametrizations of isogeny-torsion graphs of elliptic curves over $\mathbb Q$
Presenters
Abstract
The isogeny graph of a $\mathbb Q$-isogeny class of elliptic curves defined over $\mathbb Q$ consists in a vertex for each elliptic curve in the isogeny class and an edge for each rational isogeny of prime degree between elliptic curves in the isogeny class, with the degree recorded as a label on the edge. The isogeny graphs of elliptic curves over $\mathbb Q$ first appeared in the so-called Antwerp tables. Although the first proof (in press) seems to be due to Chiloyan and Lozano-Robledo in 2021. In the first part of this talk we will show parametrizations of these isogeny graphs. Moreover, Chiloyan and Lozano-Robledo define isogeny-torsion graph to be an isogeny graph where, in addition, each vertex is labeled with the abstract group structure of the torsion subgroup of the corresponding elliptic curve. They classify all the possible isogeny-torsion graphs that occur for $\mathbb Q$-isogeny classes of elliptic curves defined over $\mathbb Q$. In the last part of this talk we will show parametrizations of these isogeny-torsion graphs.
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