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Let $X_0$ be a smooth geometrically connected variety defined over a finitefield $\mathbb F_q$ and let $\mathcal E_0^{\dagger}$ be an irreducibleoverconvergent $F$-isocrystal on $X_0$. We show that if a subobject of minimalslope of the associated convergent $F$-isocrystal $\mathcal E_0$ admits anon-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 anoverconvergent isocrystal. This proves a special case of a conjecture ofKedlaya. 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 newinput in this setting is that the subgroup contains a maximal torus of theentire monodromy group. This is a consequence of the existence of a Frobeniustorus of maximal dimension. As an application, we prove a finiteness result forthe torsion points of abelian varieties, which extends the previous theorem ofLang--N\'eron and answers positively a question of Esnault.

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