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 Dieudonné modules of abelian varieties defined over global function fields of characteristic p . As a corollary we deduce that monodromy groups of such overconvergent crystalline Dieudonné modules are reductive, and after a finite base change of coefficients their connected components are the same as the connected components of monodromy groups of Galois representations on the corresponding l -adic Tate modules, for l different from p . We also show such a result for general compatible systems incorporating overconvergent F -isocrystals, conditional on a result of Abe.