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

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–N'eron and answers positively a question of Esnault.