Constancy of Newton polygons of $F$-isocrystals on Abelian varieties and isotriviality of families of curves

We prove constancy of Newton polygons of all convergent $F$-isocrystals on Abelian varieties over finite fields. Applying the constancy, we prove the isotriviality of projective smooth families of curves over Abelian varieties. We also study the isotriviality over simply connected projective smooth varieties.