  We prove an analogue of the Tate isogeny conjecture and the semi-simplicity
conjecture for overconvergent crystalline Dieudonn\'e modules of abelian
varieties defined over global function fields of characteristic $p$. As a
corollary we deduce that monodromy groups of such overconvergent crystalline
Dieudonn\'e 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.
