Locally Symmetric Families of Curves and Jacobians

In this paper we study locally symmetric families of curves and jacobians. By a locally symmetric family of jacobians, we mean a family of jacobians parameterized by a locally symmetric variety where the image of the period map is a locally symmetric subvariety of Ag[l], the moduli space of principally polarized abelian varieties of dimension g with a level l structure. A locally symmetric family of curves is a family of stable curves over a locally symmetric variety where Pic of each curve in the family is an abelian variety and where the corresponding family of jacobians is a locally symmetric family. (We’ll always fix a level l ≥ 4 to guarantee that all the arithmetic and mapping class groups we are dealing with are torsion free and the corresponding varieties are smooth.) Not every locally symmetric family of jacobians can be lifted to a locally symmetric family of curves, even if one passes to arbitrary finite covers of the base, as can be seen by looking at the universal abelian variety of dimension 3.