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Kummer Surfaces, Isogenies and Theta Functions

Adrian Clingher, Andreas Malmendier, Tony Shaska

TL;DR

The paper advances the study of $(n,n)$-isogenies for genus-2 Jacobians by integrating a Theta-function framework with explicit Kummer-surface models (Hudson, Göpel, Rosenhain) and by extending classical Richelot theory to higher odd $n$ via the Lubicz–Robert approach, achieving $O(n^2)$ complexity. It develops a full geometric and computational pipeline: from Abelian-variety theory and Theta-constants to concrete isogeny constructions on Kummer surfaces, including Shioda–Inose connections and the Dolgachev–Lehavi embedding for higher-degree kernels. The work also analyzes the Humbert loci $H_{n^2}$ and the loci $\mathcal{L}_n$ of $(n,n)$-Split Jacobians, linking algebraic geometry with cryptographic considerations, and discusses practical implications for isogeny-based cryptography, including recent attack models (Costello–Santos) and AI-assisted detection of split loci. Overall, the paper provides both a rigorous geometric framework and practical algorithms for computing and analyzing $(n,n)$-isogenies on genus-2 Jacobians, with significant implications for security and efficiency in genus-2 cryptography.

Abstract

The paper discusses geometric and computational aspects associated with $(n,n)$-isogenies for principally polarized Abelian surfaces and related Kummer surfaces. We start by reviewing the comprehensive Theta function framework for classifying genus-two curves, their principally polarized Jacobians, as well as for establishing explicit quartic normal forms for associated Kummer surfaces. This framework is then used for practical isogeny computations. A particular focus of the discussion is the $(n,n)$-Split isogeny case. We also explore possible extensions of Richelot's $(2,2)$-isogenies to higher order cases, with a view towards developing efficient isogeny computation algorithms.

Kummer Surfaces, Isogenies and Theta Functions

TL;DR

The paper advances the study of -isogenies for genus-2 Jacobians by integrating a Theta-function framework with explicit Kummer-surface models (Hudson, Göpel, Rosenhain) and by extending classical Richelot theory to higher odd via the Lubicz–Robert approach, achieving complexity. It develops a full geometric and computational pipeline: from Abelian-variety theory and Theta-constants to concrete isogeny constructions on Kummer surfaces, including Shioda–Inose connections and the Dolgachev–Lehavi embedding for higher-degree kernels. The work also analyzes the Humbert loci and the loci of -Split Jacobians, linking algebraic geometry with cryptographic considerations, and discusses practical implications for isogeny-based cryptography, including recent attack models (Costello–Santos) and AI-assisted detection of split loci. Overall, the paper provides both a rigorous geometric framework and practical algorithms for computing and analyzing -isogenies on genus-2 Jacobians, with significant implications for security and efficiency in genus-2 cryptography.

Abstract

The paper discusses geometric and computational aspects associated with -isogenies for principally polarized Abelian surfaces and related Kummer surfaces. We start by reviewing the comprehensive Theta function framework for classifying genus-two curves, their principally polarized Jacobians, as well as for establishing explicit quartic normal forms for associated Kummer surfaces. This framework is then used for practical isogeny computations. A particular focus of the discussion is the -Split isogeny case. We also explore possible extensions of Richelot's -isogenies to higher order cases, with a view towards developing efficient isogeny computation algorithms.

Paper Structure

This paper contains 31 sections, 44 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Abelian varieties
  3. Isogenies of Abelian varieties
  4. Theta functions
  5. Jacobian Varieties
  6. Endomorphism rings of Abelian varieties
  7. Genus 2 Curves and their Jacobians
  8. Principal polarizations and Torelli theorem
  9. Göpel groups and Theta divisors
  10. Theta constants of genus two
  11. Richelot isogenies
  12. An explicit model using Theta constants
  13. Kummer Surfaces
  14. Kummer surfaces with principal polarizations
  15. Kummer surfaces from odd characteristics
  16. ...and 16 more sections

Key Result

Lemma 1

If there exists an algorithm to compute $\mathop{\mathrm{End }}\nolimits_K (\mathcal{A})$ for any Abelian variety of dimension $g \geq 1$ defined over a number field $K$, then there is an algorithm to compute $\mathop{\mathrm{End }}\nolimits_{\bar{K}} (\mathcal{A})$.

Theorems & Definitions (82)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1: Poincaré-Weil
  • proof
  • Corollary 1
  • proof
  • Lemma 3
  • proof
  • ...and 72 more