Given a family of Galois coverings of the projective line, we give a simple sufficient condition ensuring that the closure of the image of the family via the period mapping is a special (or Shimura) subvariety of Ag. By a computer program we get the list of all families in genus g ≤ 9 satisfying our condition. There are no families with g = 8, 9, all of them are in genus g ≤ 7. These examples are related to a conjecture of Oort. Among them we get the cyclic examples constructed by various authors (Shimura, Mostow, De Jong-Noot, Rohde, Moonen and others) and the abelian non-cyclic examples found by Moonen-Oort. We get 7 new non-abelian examples.

Abstract. We study submanifolds of Ag that are totally geodesic for the locally symmetric metric and which are contained in the closure of the Jacobian locus but not in its boundary. In the first section we recall a formula for the second fundamental form of the period map Mg ֒→ Ag due to Pirola, Tortora and the first author. We show that this result can be stated quite neatly using a line bundle over the product of the curve with itself. We give an upper bound for the dimension of a germ of a totally geodesic submanifold passing through [C] ∈ Mg in terms of the gonality of C. This yields an upper bound for the dimension of a germ of a totally geodesic submanifold contained in the Jacobian locus, which only depends on the genus. We also study the submanifolds of Ag obtained from cyclic covers of P 1 . These have been studied by various authors. Moonen determined which of them are Shimura varieties using deep results in positive characteristic. Using our methods we show that many of the submanifolds which are not Shimura varieties are not even totally geodesic.

Abstract. We study Shimura curves of PEL type in Ag generically contained in the Prym locus. We study both the unramified Prym locus, obtained usingétale double covers, and the ramified Prym locus, corresponding to double covers ramified at two points. In both cases we consider the family of all double covers compatible with a fixed group action on the base curve. We restrict to the case where the family is 1-dimensional and the quotient of the base curve by the group is P 1 . We give a simple criterion for the image of these families under the Prym map to be a Shimura curve. Using computer algebra we check all the examples gotten in this way up to genus 28. We obtain 44 Shimura curves generically contained in the unramified Prym locus and 9 families generically contained in the ramified Prym locus. Most of these curves are not generically contained in the Jacobian locus.

Abstract. We study Shimura subvarieties of Ag obtained from families of Galois coverings f : C → C ′ where C ′ is a smooth complex projective curve of genus g ′ ≥ 1 and g = g(C). We give the complete list of all such families that satisfy a simple sufficient condition that ensures that the closure of the image of the family via the Torelli map yields a Shimura subvariety of Ag for g ′ = 1, 2 and for all g ≥ 2, 4 and for g ′ > 2 and g ≤ 9. In [13] similar computations were done in the case g ′ = 0. Here we find 6 families of Galois coverings, all with g ′ = 1 and g = 2, 3, 4 and we show that these are the only families with g ′ = 1 satisfying this sufficient condition. We show that among these examples two families yield new Shimura subvarieties of Ag, while the other examples arise from certain Shimura subvarieties of Ag already obtained as families of Galois coverings of P 1 in [13]. Finally we prove that if a family satisfies this sufficient condition with g ′ ≥ 1, then g ≤ 6g ′ + 1.

Abstract. We study the curvature of the moduli space M g of curves of genus g with the Siegel metric induced by the period map j : M g → A g . We give an explicit formula for the holomorphic sectional curvature of M g along a Schiffer variation ξ P , for P a point on the curve X, in terms of the holomorphic sectional curvature of A g and the second Gaussian map. Finally we extend the Kähler form of the Siegel metric as a closed current on M g and we determine its cohomology class as a multiple of λ.