The monodromy groups of 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.

37 pages; to appear in Selecta Mathematica

References in corpus (2)

• Independence of ell of Monodromy Groups
• Independence of l in Lafforgue's theorem

Cited by in corpus (10)

• Bessel $F$-isocrystals for reductive groups
• Etale and crystalline companions, I
• Parabolicity conjecture of $F$-isocrystals
• On the universal extensions in Tannakian categories
• Maximal tori of monodromy groups of $F$-isocrystals and an application to abelian varieties
• On the ordinary Hecke orbit conjecture
• Monodromy and Irreducibility of Igusa Varieties
• A crystalline incarnation of Berthelot's conjecture and Künneth formula for isocrystals
• Some remarks on the companions conjecture for normal varieties
• Slopes of $F$-isocrystals over abelian varieties