2-Adic Galois images of isogeny-torsion graphs over ℚ with CM

Abstract

Let [Formula: see text] be a [Formula: see text]-isogeny class of elliptic curves defined over [Formula: see text]. The isogeny graph associated to [Formula: see text] is a graph which has a vertex for each elliptic curve over [Formula: see text] of [Formula: see text] and an edge for each [Formula: see text]-isogeny of prime degree that maps one elliptic curve in [Formula: see text] to another elliptic curve in [Formula: see text], with the degree of the isogeny recorded as a label of the edge. The isogeny-torsion graph associated to [Formula: see text] is the isogeny graph associated to [Formula: see text] where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over [Formula: see text] of the corresponding elliptic curve. The main result of the article is a classification of the [Formula: see text]-adic Galois image at each vertex of the isogeny-torsion graphs whose associated [Formula: see text]-isogeny class consists of elliptic curves over [Formula: see text] with complex multiplication.