Abstract: To an algebraic curve C one can associate its Jacobian Jac(C), as was done by Abel, Jacobi, Riemann and by Weil (1948) over an arbitrary base field. Torelli proved (1913) that an isomorphism between Jac(C) and Jac(D) determines an isomorphism between C and D. However, and that is the gist of my talk, we will see that properties of C, on the one hand, and properties of Jac(C), on the other hand, are difficult to compare. It is as if an algebraic curve, and its Jacobian variety live in different worlds. We will discuss a conjecture by Coleman (1987), see that it does not hold (A. J. de Jong and R. Noot, 1991), but that a slightly weaker form does hold (joint work with Ching-Li Chai). The construction C \mapsto Jac(C) defines a map of moduli spaces. We will discuss properties of these two spaces, and an open problem which shows how little we still understand of this beautiful field (joint work in a survey paper with Ben Moonen).
