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Pré-Publication, Document De Travail Année : 2022

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

Résumé

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 underlying convergent F-isocrystal $\mathcal E_0$ admits a non-zero morphism to $\mathcal O_{X_0}$ as convergent isocrystal, then $\mathcal E_0^{\dagger}$ is isomorphic to $\mathcal O^{\dagger}_{X_0}$ as 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-N\'eron and answers positively a question of Esnault.

Dates et versions

hal-03515502 , version 1 (06-01-2022)

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Emiliano Ambrosi, Marco d'Addezio. Maximal tori of monodromy groups of $F$-isocrystals and an application to abelian varieties. 2022. ⟨hal-03515502⟩
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