The Oort conjecture on Shimura curves in the Torelli locus of curves

Oort has conjectured that there do not exist Shimura varieties of dimension >0 contained generically in the Torelli locus of genus-g curves when g is sufficiently large. In this paper we prove the Oort conjecture for Shimura curves of Mumford type and Shimura curves parameterizing principally polarized g-dimensional abelian varieties isogenous to g-fold self-products of elliptic curves for g>11. As a consequence, we obtain a finiteness result regarding smooth genus-g curves with completely decomposable Jacobians, which is related to a question of Ekedahl and Serre.