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Abelian surfaces in Hesse form and explicit isogeny formulas

Thomas Decru, Sabrina Kunzweiler

TL;DR

The paper develops an explicit framework for computing $(3,3)$-isogenies between principally polarized abelian surfaces in Hesse form by exploiting a symmetric level-$3$ theta structure, in which the $3$-torsion acts linearly in a $\mathbb{P}^8$ embedding. The authors decompose isogeny evaluation into four core operations—$S_K$, $\odot^3$, $M_{3,3}$, and $C_\lambda$—yielding a practical recipe with nine cubings and nine scalings, and they demonstrate applicability to reducible surfaces and twists. They connect these formulas to the Burkhardt quartic and its Hessian/dual, situating the construction in a rich geometric framework and comparing it to existing level-$2$ and level-$\ell$ methodologies. The work includes a detailed example over a finite field, discusses non-generic and arbitrary-kernel cases, and outlines natural generalizations to higher dimensions and different levels, accompanied by open-source SageMath code. Overall, the approach provides an efficient, model-specific route to explicit $(3,3)$-isogenies with broad potential in number-theory computations and isogeny-based cryptography.

Abstract

We develop a new method for the computation of $(3,3)$-isogenies between principally polarized abelian surfaces. The idea is to work with models in $\mathbb{P}^8$ induced by a symmetric level-$3$ theta structure. In this setting, the action of three-torsion points is linear, and the isogeny formulas can be described in a simple way as the composition of easy-to-evaluate maps. In the description of these formulas, the relation with the Burkhardt quartic threefold plays an important role. Furthermore, we discuss generalizations of the idea to higher dimensions as well as different isogeny degrees.

Abelian surfaces in Hesse form and explicit isogeny formulas

TL;DR

The paper develops an explicit framework for computing -isogenies between principally polarized abelian surfaces in Hesse form by exploiting a symmetric level- theta structure, in which the -torsion acts linearly in a embedding. The authors decompose isogeny evaluation into four core operations—, , , and —yielding a practical recipe with nine cubings and nine scalings, and they demonstrate applicability to reducible surfaces and twists. They connect these formulas to the Burkhardt quartic and its Hessian/dual, situating the construction in a rich geometric framework and comparing it to existing level- and level- methodologies. The work includes a detailed example over a finite field, discusses non-generic and arbitrary-kernel cases, and outlines natural generalizations to higher dimensions and different levels, accompanied by open-source SageMath code. Overall, the approach provides an efficient, model-specific route to explicit -isogenies with broad potential in number-theory computations and isogeny-based cryptography.

Abstract

We develop a new method for the computation of -isogenies between principally polarized abelian surfaces. The idea is to work with models in induced by a symmetric level- theta structure. In this setting, the action of three-torsion points is linear, and the isogeny formulas can be described in a simple way as the composition of easy-to-evaluate maps. In the description of these formulas, the relation with the Burkhardt quartic threefold plays an important role. Furthermore, we discuss generalizations of the idea to higher dimensions as well as different isogeny degrees.
Paper Structure (31 sections, 21 theorems, 111 equations, 1 figure)

This paper contains 31 sections, 21 theorems, 111 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contributions
  3. Comparison with isogeny formulas in the literature
  4. Applications and perspectives
  5. Outline
  6. Acknowledgements
  7. The Burkhardt quartic
  8. Automorphism group
  9. Burkhardt dual
  10. Hessian of the Burkhardt quartic
  11. Principally polarized abelian surfaces in Hesse form
  12. The $2$-torsion
  13. Relation with the Burkhardt quartic
  14. Symplectic transformations
  15. Product surfaces
  16. ...and 16 more sections

Key Result

Lemma 2.2

The singular locus of the Burkhardt quartic, $\mathcal{B}^{\text{sing}}$, consists of $45$ singular points. Explicitly, the singularities $h = (h_0:h_1:h_2:h_3:h_4) \in \mathcal{B}$ are given by and $\sigma \in \mathcal{S}_4 \subset \mathcal{S}_5$ a permutation fixing $h_0= 0$, Moreover, $\text{Aut}(\mathcal{B})$ acts transitively on $\mathcal{B}^{\text{sing}}$.

Figures (1)

  • Figure 1: Relations between the used elements of the Burkhardt quartic, its dual and its Hessian.

Theorems & Definitions (54)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Theorem 3.1: Theorem 8.3 in gunji2006defining
  • Remark 3.2
  • Definition 3.3
  • Remark 3.4
  • Remark 3.5
  • ...and 44 more