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$F$-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 proper smooth families of curves over Abelian varieties. More generally, we prove the isotriviality over projective smooth varieties on which any convergent 
$F$-isocrystal has constant Newton polygons.
