Gluing curves of genus 1 and 2 along their 2-torsion

Abstract

Let X X (resp. Y Y ) be a curve of genus 1 1 (resp. 2 2 ) over a base field k k whose characteristic does not equal 2 2 . We give criteria for the existence of a curve Z Z over k k whose Jacobian is up to twist ( 2 , 2 , 2 ) (2,2,2) -isogenous to the products of the Jacobians of X X and Y Y . Moreover, we give algorithms to construct the curve Z Z once equations for X X and Y Y are given. The first of these is based on interpolation methods involving numerical results over C \mathbb {C} that are proved to be correct over general fields a posteriori, whereas the second involves the use of hyperplane sections of the Kummer variety of Y Y whose desingularization is isomorphic to X X . As an application, we find a twist of a Jacobian over Q \mathbb {Q} that admits a rational 70 70 -torsion point.