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Generalized class group actions on oriented elliptic curves with level structure

Sarah Arpin, Wouter Castryck, Jonathan Komada Eriksen, Gioella Lorenzon, Frederik Vercauteren

TL;DR

The paper extends the action of class groups on oriented elliptic curves to a broad family of generalized class groups attached to imaginary quadratic orders, proving a free, essentially transitive action on primitively $O$-oriented elliptic curves with compatible level structure under suitable conditions. It inverts the usual viewpoint by starting from a generalized class group and deriving the corresponding level structure, then connects to suborder class groups and provides several concrete examples. A generalized exact sequence relates these groups to classical ray-class and full class groups, and the work analyzes when such actions are transitive, including the interplay with orientations in the supersingular setting. The security discussion mirrors CSIDH-type vectorization questions, showing reductions to the standard class-group action and highlighting cases where vectorization may become easier or remain hard depending on the level structure and modulus. Overall, the framework unifies level-structure actions across maximal and non-maximal orders and clarifies how generalized class groups can govern isogeny-based constructions and their security assumptions.

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.

Generalized class group actions on oriented elliptic curves with level structure

TL;DR

The paper extends the action of class groups on oriented elliptic curves to a broad family of generalized class groups attached to imaginary quadratic orders, proving a free, essentially transitive action on primitively -oriented elliptic curves with compatible level structure under suitable conditions. It inverts the usual viewpoint by starting from a generalized class group and deriving the corresponding level structure, then connects to suborder class groups and provides several concrete examples. A generalized exact sequence relates these groups to classical ray-class and full class groups, and the work analyzes when such actions are transitive, including the interplay with orientations in the supersingular setting. The security discussion mirrors CSIDH-type vectorization questions, showing reductions to the standard class-group action and highlighting cases where vectorization may become easier or remain hard depending on the level structure and modulus. Overall, the framework unifies level-structure actions across maximal and non-maximal orders and clarifies how generalized class groups can govern isogeny-based constructions and their security assumptions.

Abstract

We study a large family of generalized class groups of imaginary quadratic orders and prove that they act freely and (essentially) transitively on the set of primitively -oriented elliptic curves over a field (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 on the set of -oriented elliptic curves, discuss several other examples, and briefly comment on the hardness of the corresponding vectorization problems.
Paper Structure (18 sections, 13 theorems, 44 equations)

This paper contains 18 sections, 13 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Elliptic Curves with Level Structure
  4. Congruence subgroups and generalized class groups
  5. Class groups of suborders
  6. Class group actions on sets of elliptic curves
  7. Generalized class group actions
  8. ${\mathfrak{m}}$-level structures
  9. A family of congruence subgroups
  10. A generalised exact sequence
  11. Suborder class group actions
  12. Further examples
  13. Cyclic torsion.
  14. Scaling by $n$-th powers.
  15. The full class group.
  16. ...and 3 more sections

Key Result

theorem 1

Let Then the map where it can be assumed that $[\mathfrak{a}]$ is represented by an integral $O'$-ideal that is coprime with $f O'$, is an isomorphism of groups.

Theorems & Definitions (38)

  • definition 1: Level structure
  • theorem 1
  • proof
  • theorem 2: kopplagarias2022
  • proof
  • theorem 3: Wat69
  • definition 2
  • definition 3
  • remark 1
  • definition 4: $K$-oriented isogeny
  • ...and 28 more