  We show that a given rational Shimura curve Y with strictly maximal Higgs
field in the moduli space of g-dimensional abelian varieties does not
generically intersect the Schottky locus for large g.
  We achieve this by using a result of Viehweg and Zuo which says that if Y
parameterizes a family of curves of genus g, then the corresponding family of
Jacobians is isogenous over Y to the g-fold product of a modular family of
elliptic curves. After reducing the situation from the field of complex numbers
to a finite field, we will see, combining the Weil and Sato-Tate conjectures,
that this is impossible for large genus g.
