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 **P**^{1} 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(*X*^{4} + *Y*^{4} + *Z*^{4})
− 96914(*X*^{2} *Y*^{2} + *X*^{2} *Z*^{2}
+ *Y*^{2} *Z*^{2}) = 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).