Infinite Families of Pairs of Curves Over Q with Isomorphic Jacobians

Abstract

We present three families of pairs of geometrically non-isomorphic curves whose Jacobians are isomorphic to one another as unpolarized abelian varieties. Each family is parametrized by an open subset of P^1. The first family consists of pairs of genus-2 curves whose equations are given by simple expressions in the parameter; the curves in this family have reducible Jacobians. The second family also consists of pairs of genus-2 curves, but generically the curves in this family have absolutely simple Jacobians. The third family consists of pairs of genus-3 curves, one member of each pair being a hyperelliptic curve and the other a plane quartic. Examples from these families show that in general it is impossible to tell from the Jacobian of a genus-2 curve over Q whether or not the curve has rational points -- or indeed whether or not it has real points. Our constructions depend on earlier joint work with Franck Leprevost and Bjorn Poonen, and on Peter Bending's explicit description of the curves of genus 2 whose Jacobians have real multiplication by Z[\sqrt{2}].