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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 P1 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 15625(X4 + Y 4 + Z4) 96914(X2Y 2 + X2Z2 + Y 2Z2) = 0, whose Jacobian has 864 rational torsion points.

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