Hi. 


What are the simplest examples to have in mind of non-isomorphic smooth (complex) algebraic curves with isomorphic jacobian variety?



Simplest complex curves with isomorphic Jacobian
It is not easy to give equations of non-isomorphic Riemann surfaces whose Jacobians are isomorphic as (unpolarized) Abelian varieties.


Some explicit examples can be found in the paper by Everett W. Howe [*Infinite Families of Pairs of Curves Over $\mathbb{Q}$ with Isomorphic Jacobians*](http://jlms.oxfordjournals.org/content/72/2/327), J. London Math. Soc. (2005) **72** (2), 327-350. 


A preprint version of the paper is also available in the arXiv website, see <http://arxiv.org/abs/math.AG/0304471>. In Section 8 you can find the examples you are looking for.


