A natural object in the study of curves is the Jacobian of a curve. What are some natural geometric properties of the curve that the Jacobian encapsulates? In other words, what can the Jacobian tell us about the curve that we didn't know already?
Note, I am asking for concrete examples, statements like "The Jacobian having property blah implies the curve has property blah."
Ideally these will be statements that are easier to prove using the Jacobian (whose construction is not so easy!) rather than directly from the curve.
(Also, if this question is more appropriate for math.se, I'd be happy to delete it.)