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
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.
This paper has appeared in Forum Math. 12 (2000) 315-364.
Everett W. Howe <firstname.lastname@example.org>
Franck Leprevost <email@example.com>
Bjorn Poonen <firstname.lastname@example.org>