Infinite Families Of Isogeny-Torsion Graphs

2021/04/02 by Chiloyan, Garen
#11G05 #14H52 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2104.01128

Abstract

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

Related

• 2-adic images of CM isogeny-torsion graphs
• 2-adic Galois images of non-CM isogeny-torsion graphs
• On the number of elliptic curves with prescribed isogeny or torsion\n group over number fields of prime degree
• Polynomial bounds on torsion from a fixed geometric isogeny class of elliptic curves
• Semisimple pointed isogeny graphs for abelian varieties