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

Howe, Everett W.; Leprevost, Franck; Poonen, Bjorn

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

Authors

Howe, Everett W.

Leprevost, Franck

Poonen, Bjorn

Description

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.