2020/01/31 by Garen Chiloyan, Álvaro Lozano-Robledo · 1 citation
Mathematics · #math.NT #msc:11G05 #msc:14H52
paper · pdf · doi:10.1112/tlm3.12024
We would like to thank the referees for their helpful comments. In this version we fixed a couple of minor things. The paper has been accepted to appear in the Transactions of the London Mathematical Society
arxiv created 2020/12/14 · arxiv updated 2021/01/27
Let E be a ℚ-isogeny class of elliptic curves defined over ℚ. The isogeny graph associated to E is a graph which has a vertex for each elliptic curve in the ℚ-isogeny class E, and an edge for each cyclic 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 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 ℚ-isogeny classes of elliptic curves defined over the rationals.