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 $\mathcal E_0^{\dagger}$ 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 $\mathcal E_0^{\dagger}$ is isomorphic to $\mathcal O^{\dagger}_{X_0}$ 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 $\mathcal E_0^{\dagger}$ 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.