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Some explicit arithmetic on curves of genus three and their applications

Tomoki Moriya, Momonari Kudo

TL;DR

This work addresses explicit arithmetic on genus $3$ curves by developing algorithms for decomposed Richelot isogenies and providing explicit equations for Howe curves in both hyperelliptic and non-hyperelliptic cases. It leverages automorphism theory and fiber-product constructions to realize codomains of completely decomposed Richelot isogenies and to construct genus $3$ Howe curves with explicit models. A major contribution is the enumeration of superspecial generalized Howe curves of genus $3$, with complexity bounds $\tilde{O}(p^3)$ to $\tilde{O}(p^4)$ and detailed computational results, including Magma implementations. The results advance practical generation and classification of genus $3$ curves with Richelot decompositions and inform isogeny-graph constructions in arithmetic geometry and cryptography.

Abstract

A Richelot isogeny between Jacobian varieties is an isogeny whose kernel is included in the $2$-torsion subgroup of the domain. A Richelot isogeny whose codomain is the product of two or more principally polarized abelian varieties is called a decomposed Richelot isogeny. In this paper, we develop some explicit arithmetic on curves of genus $3$, including algorithms to compute the codomain of a decomposed Richelot isogeny. As solutions to compute the domain of a decomposed Richelot isogeny, explicit formulae of defining equations for Howe curves of genus $3$ are also given. Using the formulae, we shall construct an algorithm with complexity $\tilde{O}(p^3)$ (resp. $\tilde{O}(p^4)$) to enumerate all hyperelliptic (resp. non-hyperelliptic) superspecial Howe curves of genus $3$.

Some explicit arithmetic on curves of genus three and their applications

TL;DR

This work addresses explicit arithmetic on genus curves by developing algorithms for decomposed Richelot isogenies and providing explicit equations for Howe curves in both hyperelliptic and non-hyperelliptic cases. It leverages automorphism theory and fiber-product constructions to realize codomains of completely decomposed Richelot isogenies and to construct genus Howe curves with explicit models. A major contribution is the enumeration of superspecial generalized Howe curves of genus , with complexity bounds to and detailed computational results, including Magma implementations. The results advance practical generation and classification of genus curves with Richelot decompositions and inform isogeny-graph constructions in arithmetic geometry and cryptography.

Abstract

A Richelot isogeny between Jacobian varieties is an isogeny whose kernel is included in the -torsion subgroup of the domain. A Richelot isogeny whose codomain is the product of two or more principally polarized abelian varieties is called a decomposed Richelot isogeny. In this paper, we develop some explicit arithmetic on curves of genus , including algorithms to compute the codomain of a decomposed Richelot isogeny. As solutions to compute the domain of a decomposed Richelot isogeny, explicit formulae of defining equations for Howe curves of genus are also given. Using the formulae, we shall construct an algorithm with complexity (resp. ) to enumerate all hyperelliptic (resp. non-hyperelliptic) superspecial Howe curves of genus .
Paper Structure (21 sections, 33 theorems, 55 equations, 5 tables, 2 algorithms)

This paper contains 21 sections, 33 theorems, 55 equations, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperelliptic curves and their isomorphisms.
  4. Automorphisms of hyperelliptic curves.
  5. Generalized Howe curves
  6. Decomposed Richelot isogenies.
  7. Computing decomposed Richelot isogenies
  8. Case of hyperelliptic curves
  9. Case of genus-3 hyperelliptic curves having completely decomposed Richelot isogenies
  10. Case of smooth plane quartics (non-hyperelliptic curves of genus 3)
  11. Explicit construction of Howe curves of genus 3
  12. Hyperelliptic case
  13. Non-hyperelliptic case
  14. Enumeration of superspecial generalized Howe curves
  15. Some lemmas for isomorphism classification
  16. ...and 6 more sections

Key Result

Theorem 1

Let $C$ be a hyperelliptic curve of genus $g \geq 2$ over $k$ with an extra involution $\sigma$. Given $C$, there is an algorithm (explicitly, Algorithm alg:decom richelot) to compute equations defining the quotient curves $C/\langle \sigma \rangle$ and $C/\langle \sigma \circ \iota_C \rangle$, whic

Theorems & Definitions (63)

  • Theorem 1: Theorem \ref{['thm:compute decom hyper']} below
  • Theorem 2: Theorems \ref{['thm:complete hyper']} and \ref{['thm:decom richelot non-hyper']} below
  • Theorem 3
  • Lemma 2.1.1: lercier2013fast
  • Proposition 2.2.1
  • proof
  • Remark 2.2.2
  • Lemma 2.2.3
  • Proposition 2.3.1: Katsura-Takashima
  • Lemma 2.3.2: cf. Katsura-Takashima
  • ...and 53 more