The p-adic monodromy group of abelian varieties over global function fields of characteristic p

We prove an analogue of the Tate isogeny conjecture and the semi-simplicity conjecture for overconvergent crystalline Dieudonne modules of abelian varieties defined over global function fields of characteristic p, combining methods of de Jong and Faltings. As a corollary we deduce that the monodromy groups of such overconvergent crystalline Dieudonne modules are reductive, and after base change to the field of complex numbers they are the same as the monodromy groups of Galois representations on the corresponding l-adic Tate modules, for l different from p.