Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the computation of endomorphism rings of abelian surfaces over finite fields

Samuele Anni, Gaetan Bisson, Annamaria Iezzi, Elisa Lorenzo García, Benjamin Wesolowski

TL;DR

The article provides a comprehensive overview of algorithmic strategies for computing End(A) for principally polarized abelian surfaces over finite fields, categorized by p-rank and simplicity. It establishes the computability of End(A) in general (theoretical existence of an algorithm) and then develops specialized, case-diverse methods: generic endomorphism testing, lifting to characteristic zero, and order-ascending techniques for simple ordinary surfaces; for non-simple surfaces it introduces two elliptic-factor approaches via coprime isogenies and elliptic subcovers, with concrete treatment of p-rank 0, 1, and 2 cases and supersingular instances using random-walk isogeny graphs. The paper also addresses abelian surfaces with extra automorphisms, providing explicit decompositions and families that facilitate End computations. Overall, it advances practical and theoretical understanding of End(A) across all genus-two surface types, linking CM-field arithmetic, isogeny theory, and lattice methods to endomorphism-ring exhaustiveness and explicit generation. The results have implications for isogeny-based cryptography and the broader computational landscape of abelian-variety arithmetic.

Abstract

We study endomorphism rings of principally polarized abelian surfaces over finite fields from a computational viewpoint with a focus on exhaustiveness. In particular, we address the cases of non-ordinary and non-simple varieties. For each possible surface type, we survey known results and, whenever possible, provide improvements and missing results.

On the computation of endomorphism rings of abelian surfaces over finite fields

TL;DR

The article provides a comprehensive overview of algorithmic strategies for computing End(A) for principally polarized abelian surfaces over finite fields, categorized by p-rank and simplicity. It establishes the computability of End(A) in general (theoretical existence of an algorithm) and then develops specialized, case-diverse methods: generic endomorphism testing, lifting to characteristic zero, and order-ascending techniques for simple ordinary surfaces; for non-simple surfaces it introduces two elliptic-factor approaches via coprime isogenies and elliptic subcovers, with concrete treatment of p-rank 0, 1, and 2 cases and supersingular instances using random-walk isogeny graphs. The paper also addresses abelian surfaces with extra automorphisms, providing explicit decompositions and families that facilitate End computations. Overall, it advances practical and theoretical understanding of End(A) across all genus-two surface types, linking CM-field arithmetic, isogeny theory, and lattice methods to endomorphism-ring exhaustiveness and explicit generation. The results have implications for isogeny-based cryptography and the broader computational landscape of abelian-variety arithmetic.

Abstract

We study endomorphism rings of principally polarized abelian surfaces over finite fields from a computational viewpoint with a focus on exhaustiveness. In particular, we address the cases of non-ordinary and non-simple varieties. For each possible surface type, we survey known results and, whenever possible, provide improvements and missing results.

Paper Structure

This paper contains 29 sections, 27 theorems, 35 equations, 1 figure, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representing abelian surfaces and their endomorphisms
  4. Computing basic invariants
  5. Computability of endomorphism rings
  6. Generic endomorphism testing
  7. Lifting to characteristic zero
  8. Simple abelian surfaces
  9. Simple, ordinary case
  10. Testing candidate orders
  11. The method of Eisenträger--Lauter eisentrager-lauter.
  12. The method of Bisson end-g2.
  13. The method of Robert Robert22c.
  14. Ascending the lattice of orders
  15. Simple, $p$-rank-$1$ case
  16. ...and 14 more sections

Key Result

Proposition 2.1

Every principally polarized abelian surface is either the Jacobian variety of a genus-two curve or the product of two elliptic curves with the product polarization.

Figures (1)

  • Figure 1: Inclusions between orders in a quartic CM-field and their intersections with the totally real subfield.

Theorems & Definitions (46)

  • Proposition 2.1: weil-torelli
  • Theorem 2.2: weil-torelli and Adleman
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Remark 3.4
  • Proposition 3.5
  • proof
  • Theorem 3.6
  • ...and 36 more