(A preprint and an official version are available online.)

We present three families of pairs of geometrically non-isomorphic
curves whose Jacobians are isomorphic to one another as unpolarized
abelian varieties. The first family consists of pairs of genus-2
curves whose equations are given by simple expressions in a single
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 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 Leprévost
and Bjorn Poonen, and on Peter Bending's
explicit description of the
curves of genus 2 whose Jacobians have real multiplication
by **Z**[√2].

An example: The curves

Another example: The Jacobian of the hyperelliptic curve

We used several Magma routines while working on this paper. They can be found here.