Gluing Genus 1 and Genus 2 Curves Along ℓ-torsion

2025/02/13 by Pitchayut Saengrungkongka, Saengrungkongka, Pitchayut, Walsh, Noah
Mathematics · #11G30 #14H25 #14H40 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2502.09753

openalex publication_date 2025/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Y be a genus 2 curve over \mathbb Q. We provide a method to systematically search for possible candidates of a prime ℓ≥ 3 and a genus 1 curve X for which there exists a genus 3 curve Z over \mathbb Q whose Jacobian is, up to quadratic twist, (ℓ, ℓ, ℓ)-isogenous to the product of Jacobians of X and Y, building on the work by Hanselman, Schiavone, and Sijsling for ℓ=2. We find several such pairs (X,Y) for prime ℓ up to 13. We also improve their numerical gluing algorithm, allowing us to successfully glue genus 1 and genus 2 curves along their 13-torsion.

Related

• Gluing curves of genus 1 and 2 along their 2-torsion
• Genus-2 Jacobians with torsion points of large order
• Computing isogenies between Jacobian of curves of genus 2 and 3
• Explicit 7-torsion in the Tate-Shafarevich groups of genus 2 Jacobians
• Some explicit arithmetic on curves of genus three and their applications