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Equivalent computational problems for superspecial abelian surfaces

Mickaël Montessinos

TL;DR

This work establishes a network of polynomial-time reductions (under GRH and KLPT^2 assumptions) among core computational problems for PPSAS: computing the Ibukiyama-Katsura-Oort matrix, obtaining an explicit endomorphism-basis representation, and finding unpolarised isomorphisms between PPSAS. By treating products of supersingular elliptic curves separately from genus 2 Jacobians, it shows that, for products, the three problems are equivalent, while for Jacobians the IKO-matrix problem corresponds to obtaining a good endomorphism representation; in all cases, End(A) computations can be reduced to IKO data and vice versa. A central technical contribution is a pair of reductions: from unpolarised isomorphisms to IKO matrices, and from IKO matrices to endomorphism rings with Rosati-tracking, implemented via chains of $(2,2)$-isogenies and KLPT^2, with explicit handling of orientations to ensure correctness of the induced Rosati involution. The results unify multiple representations of endomorphism data and open pathways to practical, though computationally intensive, procedures for recovering endomorphism rings and isomorphisms of PPSAS in the high-dimensional setting. These equivalences have potential implications for isogeny-based cryptography and the analysis of higher-dimensional endomorphism structures.

Abstract

We show reductions and equivalences between various problems related to the computation of the endomorphism ring of principally polarised superspecial abelian surfaces. Problems considered are the computation of the Ibukiyama-Katsura-Oort matrix and computation of unpolarised isomoprhisms between superspecial abelian surfaces.

Equivalent computational problems for superspecial abelian surfaces

TL;DR

This work establishes a network of polynomial-time reductions (under GRH and KLPT^2 assumptions) among core computational problems for PPSAS: computing the Ibukiyama-Katsura-Oort matrix, obtaining an explicit endomorphism-basis representation, and finding unpolarised isomorphisms between PPSAS. By treating products of supersingular elliptic curves separately from genus 2 Jacobians, it shows that, for products, the three problems are equivalent, while for Jacobians the IKO-matrix problem corresponds to obtaining a good endomorphism representation; in all cases, End(A) computations can be reduced to IKO data and vice versa. A central technical contribution is a pair of reductions: from unpolarised isomorphisms to IKO matrices, and from IKO matrices to endomorphism rings with Rosati-tracking, implemented via chains of -isogenies and KLPT^2, with explicit handling of orientations to ensure correctness of the induced Rosati involution. The results unify multiple representations of endomorphism data and open pathways to practical, though computationally intensive, procedures for recovering endomorphism rings and isomorphisms of PPSAS in the high-dimensional setting. These equivalences have potential implications for isogeny-based cryptography and the analysis of higher-dimensional endomorphism structures.

Abstract

We show reductions and equivalences between various problems related to the computation of the endomorphism ring of principally polarised superspecial abelian surfaces. Problems considered are the computation of the Ibukiyama-Katsura-Oort matrix and computation of unpolarised isomoprhisms between superspecial abelian surfaces.
Paper Structure (28 sections, 25 theorems, 19 equations, 2 algorithms)

This paper contains 28 sections, 25 theorems, 19 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Methods
  4. Concurrent works
  5. Structure of this work
  6. Open problems
  7. Notations
  8. Acknowledgments
  9. Preliminaries
  10. Principally polarised superspecial abelian surfaces
  11. Polarisations and the Rosati involution
  12. Superspecial abelian varieties
  13. The Ibukiyama-Katsura-Oort classification
  14. Computational Representations
  15. Principally polarised superspecial abelian surfaces
  16. ...and 13 more sections

Key Result

Proposition 2.7

jordan2018abelian Let $A$ be a maximal abelian variety. Then, we have the following: Furthermore, every superspecial abelian variety defined over $\overline{k}$ admits a maximal model defined over $k$.

Theorems & Definitions (64)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Lemma 2.8
  • proof
  • Proposition 2.9: castryck2025klpt
  • ...and 54 more