Everett W. Howe, Franck Leprévost, and Bjorn Poonen: Large torsion subgroups of split Jacobians of curves of genus two or three, Forum Math. 12 (2000) 315–364. MR 2001e:11071, Zbl 0983.11037.

(An official and an unofficial version are available online.)

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, we show that the rational points on a certain positive-rank elliptic surface over P1 parameterize a family of genus-2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find a genus-3 curve — namely, the curve

15625(X4 + Y4 + Z4) − 96914(X2 Y2 + X2 Z2 + Y2 Z2) = 0
— whose Jacobian has 864 rational torsion points. If your Web browser can handle tables, you can see complete lists of the torsion groups we can obtain for genus-2 Jacobians and for genus-3 Jacobians. If your Web browser can't handle tables, you can check out the genus-2 list (plain format) and the the genus-3 list (plain format).