Everett W. Howe: Constructing distinct curves with isomorphic Jacobians, J. Number Theory 56 (1996) 381–390, MR 97d:11101, Zbl 0842.14019.
(An official electronic version is available.)
I give a method for searching for explicit examples of geometrically non-isomorphic curves over a finite field whose Jacobians are isomorphic to one another (as unpolarized abelian varieties) and are absolutely simple. Several such pairs of curves, of genus two, are given. In addition, I find a hyperelliptic curve of genus three and a plane quartic (both over the field with three elements) whose Jacobians are isomorphic. Thus, one cannot determine whether or not a curve is hyperelliptic simply by looking at its unpolarized Jacobian.