Maximal tori of monodromy groups of $F$-isocrystals and an application to abelian varieties

Abstract

Let $X_0$ be a smooth geometrically connected variety defined over a finite field ${\mathbb{F}}_q$ and let ${\mathcal{E}}_0^{†}$ be an irreducible overconvergent $F$-isocrystal on $X_0$. We show that if a subobject of minimal slope of the associated convergent $F$-isocrystal ${\mathcal{E}}_0$ admits a non-zero morphism to ${\mathcal{O}}_{X_0}$ as a convergent isocrystal, then ${\mathcal{E}}_0^{†}$ is isomorphic to ${\mathcal{O}}_{X_0}^{†}$ as an overconvergent isocrystal. This proves a special case of a conjecture of Kedlaya. The key ingredient in the proof is the study of the monodromy group of ${\mathcal{E}}_0^{†}$ and the subgroup defined by ${\mathcal{E}}_0$. The new input in this setting is that the subgroup contains a maximal torus of the entire monodromy group. This is a consequence of the existence of a Frobenius torus of maximal dimension. As an application, we prove a finiteness result for the torsion points of abelian varieties, which extends the previous theorem of Lang--Néron and answers positively a question of Esnault.

19 pages; to appear in Algebraic Geometry

References in corpus (9)

• Relative log convergent cohomology and relative rigid cohomology I
• The monodromy groups of lisse sheaves and overconvergent $F$-isocrystals
• Etale and crystalline companions, I
• Parabolicity conjecture of $F$-isocrystals
• On the universal extensions in Tannakian categories
• The monodromy of $F$-isocrystals with log-decay
• Relative log convergent cohomology and relative rigid cohomology II
• Crystalline Chebotarëv density theorems
• Minimal slope conjecture of $F$-isocrystals

Cited by in corpus (5)

• Parabolicity conjecture of $F$-isocrystals
• On the universal extensions in Tannakian categories
• Boundedness of the $p$-primary torsion of the Brauer group of an abelian variety
• Perfect points of abelian varieties
• Slopes of $F$-isocrystals over abelian varieties