2026/08/03 by Alexander J. Barrios, Enrique González-Jiménez, Ivan Novak
Mathematics · #math.NT #math.AG #math.CO #msc:11G05 #msc:11G07 #msc:11G15 #msc:11F80 #msc:14K02 #msc:05C25 #msc:05C51
101 pages
arxiv created 2026/08/03 · arxiv updated 2026/08/04
For an elliptic curve E defined over a field K of characteristic 0 with EndK E ≅ 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 Hpkr and Hp∞,+r, every member of which occurs as a p-primary isogeny graph except for H2k0 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 \operatorname*GL\nolimits2(ℤp). More generally, we identify subgroups of \operatorname*GL\nolimits2(\widehatℤ) 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 \mathfrakIp(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.