Everett W. Howe: Constructing distinct curves with isomorphic Jacobians in characteristic zero, Internat. Math. Res. Notices 1995 173–180, MR 96f:14030, Zbl 0832.14019.
(An official version is available online.)
In the 1960s, Hayashida and Nishi showed that there are arbitrarily large sets of non-isomorphic curves over the complex numbers, all of whose Jacobians are isomorphic to one another (as unpolarized abelian varieties). I show how explicit examples of such sets can be constructed.