• Corpus ID: 231627731

# Slopes of $F$-isocrystals over abelian varieties

@inproceedings{dAddezio2021SlopesO,
title={Slopes of \$F\$-isocrystals over abelian varieties},
year={2021}
}
We prove that an F -isocrystal over an abelian variety defined over a perfect field of positive characteristic has constant slopes. This recovers and extends a theorem of Tsuzuki for abelian varieties over finite fields. Our proof exploits the theory of monodromy groups of convergent isocrystals.
1 Citations
A crystalline incarnation of Berthelot’s conjecture and Künneth formula for isocrystals
• Mathematics
Journal of Algebraic Geometry
• 2022
Berthelot’s conjecture predicts that under a proper and smooth morphism of schemes in characteristic p p , the higher direct images of an overconvergent F F -isocrystal are

## References

SHOWING 1-10 OF 19 REFERENCES
Pentes en cohomologie rigide et F-isocristaux unipotents
• Mathematics
• 1999
Abstract:We study the slopes of Frobenius on the rigid cohomology and the rigid cohomology with compact support of an algebraic variety over a perfect field of positive characteristic. We then prove
The $p$-adic monodromy group of abelian varieties over global function fields of characteristic $p$
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
CONSTANCY OF NEWTON POLYGONS OF $F$-ISOCRYSTALS ON ABELIAN VARIETIES AND ISOTRIVIALITY OF FAMILIES OF CURVES
• N. Tsuzuki
• Mathematics
Journal of the Institute of Mathematics of Jussieu
• 2019
Abstract We prove constancy of Newton polygons of all convergent $F$-isocrystals on Abelian varieties over finite fields. Applying the constancy, we prove the isotriviality of proper smooth families
Abelianization of the F-divided fundamental group scheme
• Mathematics
• 2016
Let (X , x0) be a pointed smooth proper variety defined over an algebraically closed field. The Albanese morphism for (X , x0) produces a homomorphism from the abelianization of the F-divided
The monodromy groups of lisse sheaves and overconvergent F-isocrystals
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
A crystalline incarnation of Berthelot’s conjecture and Künneth formula for isocrystals
• Mathematics
Journal of Algebraic Geometry
• 2022
Berthelot’s conjecture predicts that under a proper and smooth morphism of schemes in characteristic p p , the higher direct images of an overconvergent F F -isocrystal are
Fourier transforms and $p$-adic ‘Weil II’
We give a purity theorem in the manner of Deligne's ‘Weil II’ theorem for rigid cohomology with coefficients in an overconvergent $F$-isocrystal; the proof mostly follows Laumon's Fourier-theoretic
On higher direct images of convergent isocrystals
• Daxin Xu
• Mathematics
Compositio Mathematica
• 2019
Let $k$ be a perfect field of characteristic $p>0$ and let $\operatorname{W}$ be the ring of Witt vectors of $k$ . In this article, we give a new proof of the Frobenius descent for convergent
On the S-fundamental group scheme. II
• A. Langer
• Mathematics
Journal of the Institute of Mathematics of Jussieu
• 2012
Abstract The S-fundamental group scheme is the group scheme corresponding to the Tannaka category of numerically flat vector bundles. We use determinant line bundles to prove that the S-fundamental
Maximal tori of monodromy groups of $F$-isocrystals and an application to abelian varieties
• Mathematics
• 2018
Let $X_0$ be a smooth geometrically connected variety defined over a finite field $\mathbb F_q$ and let $\mathcal E_0^{\dagger}$ be an irreducible overconvergent $F$-isocrystal on $X_0$. We show that