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

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