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 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(X + Y 4 + Z)− 96914...

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 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)− 969...

Authors are grateful to ALISTORE‐ERI colleagues for helpful discussions. ICMAB‐CSIC members thank the Spanish Ministry Economy, Industry and Competitiveness Severo Ochoa Programme Centres of Excellence in R&D (CEX2019‐000917‐S) funding through grant MAT2017‐86616‐R. MCC is also PID2019‐107106RB‐C33.