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

Let X (resp. Y) be a curve of genus 1 (resp. 2) over a base field k whose characteristic does not equal 2. We give criteria for the existence of a curve Z over k whose Jacobian is up to twist (2,2,2)-isogenous to the products of the Jacobians of X and Y. Moreover, we give algorithms to construct the curve Z once equations for X and Y are given. The first of these is based on interpolation methods involving numerical results over 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 whose desingularization is isomorphic to X. As an application, we find a twist of a Jacobian over Q that admits a rational 70-torsion point. [ABSTRACT FROM AUTHOR]