The monodromy groups of lisse sheaves and overconvergent $F$-isocrystals

TL;DR

This paper extends the understanding of monodromy groups for compatible systems of lisse sheaves and overconvergent $F$-isocrystals on smooth varieties, generalizing previous results from curves to higher dimensions.

Contribution

It generalizes known results about monodromy groups from curves to arbitrary smooth varieties and extends theorems on Frobenius tori to overconvergent $F$-isocrystals.

Findings

Monodromy groups of compatible systems have the same connected components and neutral parts.

Extended Serre and Chin's theorem on Frobenius tori to overconvergent $F$-isocrystals.

Used Tannakian formalism to compare lisse sheaves and overconvergent $F$-isocrystals.

Abstract

It has been proven by Serre, Larsen-Pink and Chin, that over a smooth curve over a finite field, the monodromy groups of compatible semi-simple pure lisse sheaves have "the same" $\pi_0$ and neutral component. We generalize their results to compatible systems of semi-simple lisse sheaves and overconvergent $F$-isocrystals over arbitrary smooth varieties. For this purpose, we extend the theorem of Serre and Chin on Frobenius tori to overconvergent $F$-isocrystals. To put our results into perspective, we briefly survey recent developments of the theory of lisse sheaves and overconvergent $F$-isocrystals. We use the Tannakian formalism to make explicit the similarities between the two types of coefficient objects.