A bound on the dimension of a totally geodesic submanifold in the Prym locus

We give an upper bound for the dimension of a germ of a totally geodesic submanifold, and hence of a Shimura variety of Ag−1 , contained in the Prym locus. First we give such a bound for a germ passing through a Prym variety of a k-gonal curve in terms of the gonality k. Then we deduce a bound only depending on the genus g.