Abstract
For an elliptic curve E defined over a field K of characteristic 0 with EndKE≅Z, we classify which isogeny graphs G(E/K) can occur. We first show that G(E/K) decomposes as a weak Cartesian product of its p-primary isogeny graphs, one for each prime p, thereby reducing the problem to classifying p-primary isogeny graphs. We then show that each such graph is isomorphic, as an edge-weighted graph, to a member of an explicit family of edge-weighted graphs Hrpk and Hrp∞,+, every member of which occurs as a p-primary isogeny graph except for H02k for k≥2. The proof relies on a detailed study of the p-adic Galois representation attached to E, through which we identify each graph with a subgroup of GL2(Zp).
More generally, we identify subgroups of GL2(Zˆ) for each possible isogeny graph and describe their corresponding modular curves, completing, in the genus 0 case, the explicit parameterization of isogeny graphs via parameterized isogenous families of elliptic curves. We also introduce the p-blooming invariant Ip(E/K), an isogeny class invariant determining the value of r in the p-primary isogeny graph, and show that elliptic curves over fields with a real embedding attain the smallest possible value. As applications, we characterize the isogeny graphs of elliptic curves with potential complex multiplication; give an algorithm for determining the isogeny graph from the adelic Galois representation; recover the classification of rational isogeny graphs; and, under GRH, classify the isogeny graphs occurring over certain number fields.