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 . 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 -adic Tate modules, for different from . We also show such a result for general compatible systems incorporating overconvergent -isocrystals, conditional on a result of Abe.
Cite this article
Ambrus Pál, The -adic monodromy group of abelian varieties over global function fields of characteristic . Doc. Math. 27 (2022), pp. 1509–1579DOI 10.4171/DM/903