Generalized class group actions on oriented elliptic curves with level structure

TL;DR

This paper generalizes the action of class groups on oriented elliptic curves with level structure, extending previous results and exploring implications for vectorization problems.

Contribution

It introduces a broad family of class groups acting on oriented elliptic curves, generalizing prior work and analyzing their properties and applications.

Findings

Class groups act freely and transitively on oriented elliptic curves with level structure.

Extension of previous results to more general class groups and orders.

Discussion on the hardness of vectorization problems related to these actions.

Abstract

We study a large family of generalized class groups of imaginary quadratic orders $O$ and prove that they act freely and (essentially) transitively on the set of primitively $O$-oriented elliptic curves over a field $k$ (assuming this set is non-empty) equipped with appropriate level structure. This extends, in several ways, a recent observation due to Galbraith, Perrin and Voloch for the ray class group. We show that this leads to a reinterpretation of the action of the class group of a suborder $O' \subseteq O$ on the set of $O'$-oriented elliptic curves, discuss several other examples, and briefly comment on the hardness of the corresponding vectorization problems.