Result: Computing isogenies from modular equations in genus two

Title:
Computing isogenies from modular equations in genus two
Authors:
Jean Kieffer, Aurel Page, Damien Robert
Contributors:
Kieffer, Jean
Source:
Journal of Algebra. 666:331-386
Publication Status:
Preprint
Publisher Information:
Elsevier BV, 2025.
Publication Year:
2025
Subject Terms:
Algorithm, Mathematics - Algebraic Geometry, Mathematics - Number Theory, Modular equations, Isogenies, [MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG], FOS: Mathematics, Number Theory (math.NT), Abelian surfaces, 0101 mathematics, 01 natural sciences, Algebraic Geometry (math.AG), [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]
Document Type:
Academic journal Article
File Description:
application/pdf
Language:
English
ISSN:
0021-8693
DOI:
10.1016/j.jalgebra.2024.11.029
DOI:
10.48550/arxiv.2001.04137
Access URL:
http://arxiv.org/abs/2001.04137
https://hal.science/hal-02436133v3/document
https://hal.science/hal-02436133v3
https://doi.org/10.1016/j.jalgebra.2024.11.029
Rights:
Elsevier TDM
arXiv Non-Exclusive Distribution
Accession Number:
edsair.doi.dedup.....ff4601e4ed6db61cfbc6afc78bfc12f9
Database:
OpenAIRE

Further Information

We present an algorithm solving the following problem: given two genus 2 curves over a field k with isogenous Jacobians, compute such an isogeny explicitly. This isogeny can be either an l-isogeny or, in the real multiplication case, an isogeny with cyclic kernel; we require that k have large enough characteristic and that the curves be sufficiently generic. Our algorithm uses modular equations for these isogeny types, and makes essential use of an explicit Kodaira--Spencer isomorphism in genus 2.