Families of Pairs of Genus 2 Curves with Isomorphic Unpolarized Jacobians

TL;DR

This paper constructs three families of genus 2 curve pairs over a field K with isomorphic unpolarized Jacobians, parameterized by open subsets of the projective line, and discusses obstructions to similar constructions over Q.

Contribution

It introduces explicit families of genus 2 curves with isomorphic Jacobians and analyzes limitations of the construction over the rational numbers.

Findings

Three explicit families of genus 2 curve pairs with isomorphic Jacobians

Parameterization of families by open subsets of the projective line

Identification of obstructions to similar constructions over Q

Abstract

We construct three families of pairs of genus 2 curves over a field K, whose Jacobians are isomorphic as unpolarized abelian varieties. Each family is parameterized by an open subset of the Projective line over K. Our construction is based on a remark in a paper by Howe. Later, we describe an obstruction to perusing the same technique over Q.