Large torsion subgroups of split Jacobians of curves of genus two or three

We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P 1 of positive rank parameterize a family of genus-2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find the genus-3 curve