Geodesic
The $p$-adic monodromy group of abelian varieties over global function fields of characteristic $p$
Documenta mathematica, Tome 27 (2022), pp. 1509-1579.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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.
Classification : 14K15, 14F30, 14F35
Keywords: \(F\)-isocrystals, monodromy, overconvergent crystalline Dieudonné modules 
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     author = {P\'al, Ambrus},
     title = {The \(p\)-adic monodromy group of abelian varieties over global function fields of characteristic \(p\)},
     journal = {Documenta mathematica},
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     url = {http://geodesic.mathdoc.fr/item/DOCMA_2022__27__a28/}
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Pál, Ambrus. The \(p\)-adic monodromy group of abelian varieties over global function fields of characteristic \(p\). Documenta mathematica, Tome 27 (2022), pp. 1509-1579. http://geodesic.mathdoc.fr/item/DOCMA_2022__27__a28/