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

TL;DR

This paper establishes analogues of key conjectures for overconvergent crystalline Dieudonné modules of abelian varieties over global function fields of characteristic p, linking monodromy groups to Galois representations and proving their reductiveness.

Contribution

It proves the Tate isogeny and semi-simplicity conjectures for overconvergent crystalline Dieudonné modules in this setting, connecting monodromy groups with Galois representations.

Findings

Monodromy groups are reductive.

Connected components match those of Galois representations after base change.

Results extend to compatible systems with conditional assumptions.

Abstract

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.