## Finite local monodromy overconvergent unit-root $$F$$-isocrystals on a curve.(English)Zbl 0943.14007

Let $$X$$ be a smooth curve over a perfect field $$k$$ of characteristic $$p>0$$, with smooth compactification $$\overline{X}$$. Write $$D=\overline{X}-X$$ for the complement. It is shown that there is a canonical equivalence between the category of $$p$$-adic representations of the fundamental group $$\pi_1(X,*)$$ with finite local monodromy and the category of overconvergent unit-root $$F$$-isocrystals on $$X$$, thus generalizing the rank one case of R. Crew [in: Algebraic Geometry, Proc. Summer Res. Inst., Brunswick 1985, Part 2, Proc. Symp. Pure Math. 46, 111-138 (1987; Zbl 0639.14011)]. Finite local monodromy means that, for any closed point $$x\in D$$, the inertia subgroup at $$x$$ acts through a finite quotient. The method of proof is to construct a local theory and then apply it to the global problem.
Let $$\Lambda$$ be a discrete valuation field which is finite over $${\mathbb Q}_p$$. Write $$q=p^f$$ for the cardinality of the residue class field of $$\Lambda$$. Assume $${\mathbb F}_q\subset k$$. Put $$K=\Lambda\otimes_{W({\mathbb F}_q)}W(k)$$, where $$W$$ denotes the ring of Witt vectors. Write $$|\;|$$ for the absolute value on the $$p$$-adic completion $$\widehat{K}^{\text{alg}}$$ of the algebraic closure $$K^{\text{alg}}$$ of $$K$$. Then for an extension $$L$$ of $$K$$ in $$\widehat{K}^{\text{alg}}$$ one defines $$L$$-algebras $${\mathcal E}_{t,L}=\{\sum_{n=-\infty}^{n=\infty}a_nt^n\mid a_n\in L$$, $$|a_n|$$ is bounded, $$|a_n|\to 0$$ $$(n\to -\infty)\}$$ and $$\mathcal E^\dagger_{t,L}=\{\sum a_nt^n\in\mathcal E_{t,L}\mid |a_n|\rho^n\to 0$$ $$(n\to -\infty)$$ for some $$0<\rho<1\}$$. One writes $$\mathcal E$$ (resp. $$\mathcal E^\dagger$$) for $$\mathcal E_{t,K}$$ (resp. $$\mathcal E^\dagger_{t,K}$$). Then $$\mathcal E^\dagger$$ is a henselian discrete valuation field. One fixes a Frobenius $$\sigma$$ on $$\mathcal E$$ and $$\mathcal E^\dagger$$. For $$R=\mathcal E$$ or $$\mathcal E^\dagger$$, one writes $$\underline{M\Phi}^\nabla_{R,\sigma}$$ (resp. $$\underline{M\Phi}^{\nabla,\text{ét}}_{R,\sigma}$$) for Fontaine’s categories of $$\varphi-\nabla$$-, resp. étale $$\varphi-\nabla$$-modules. One can define a functor $$\nu^*:\underline{M\Phi}^\nabla_{\mathcal E^\dagger,\sigma}\to\underline{M\Phi}^\nabla_{\mathcal E,\sigma}$$ by scalar extensions. One proves that $$\nu^*$$ commutes with tensor products and duals and that an object $$M\in\underline{M\Phi}^\nabla_{\mathcal E^\dagger,\sigma}$$ is étale if and only if $$\nu^*(M)$$ is étale. The functor $$\nu^*$$ is fully faithful on $$\underline{M\Phi}^{\nabla,\text{ét}}_{\mathcal E^\dagger,\sigma}$$.
Let $$F=k((t))$$ and write $$G_F$$ for $$\text{Gal}(F^{\text{sep}}/F)$$. Denote by $$\underline{\text{Rep}}_\Lambda(G_F)$$ (resp. $$\underline{\text{Rep}}_\Lambda^{\text{fin}}(G_F)$$) the category of finite dimensional $$\Lambda$$-vector spaces with a continuous $$G_F$$-action (resp. the full subcategory consisting of representations with finite monodromy, i.e. such that the inertia group $$I_F$$ acts through a finite quotient). Fontaine defined a functor $$D_\sigma:\underline{\text{Rep}}_\Lambda(G_F)\rightarrow\underline{M\Phi}^{\nabla,\text{ét}}_{\mathcal E,\sigma}$$. The functor $$D_\sigma$$ can be given explicitly. $$D_\sigma$$ turns out to be an equivalence of categories. In the underlying paper a functor $$D_\sigma^\dagger:\underline{\text{Rep}}_\Lambda^{\text{fin}}(G_F)\rightarrow\underline{M\Phi}^{\nabla,\text{ét}}_{\mathcal E^\dagger,\sigma}$$ is defined. It commutes with tensor products, duals, and inverse images, and also with $$D_\sigma$$. The main result of the paper can be stated as follows: The functor $$D^\dagger_\sigma$$ is an equivalence of categories and $$D_\sigma=\nu^*D^\dagger_\sigma$$. Moreover, $$D^\dagger_\sigma$$ commutes with tensor products, duals, and an inverse image $$f^*$$ for a finite separable extension $$f:F\rightarrow E$$ in $$F^{\text{sep}}$$. The proof of this result relies on the existence of solutions of linear ordinary Frobenius differential equations for matrices $$A\in\text{GL}_r(\mathcal E^\dagger)$$. It is quite technical.
As an application it is shown that the functor $$G^D$$ (defined by Crew) gives an equivalence between the category $$\underline{\text{Rep}}_\Lambda^D(\pi_1(U,*))$$ of continuous $$\Lambda$$-representations of the fundamental group $$\pi_1(U,*)$$, with $$U$$ a dense open subscheme of the smooth, geometrically connected curve $$X/k$$, with local finite monodromy around $$D=X-U$$ and the category of unit-root $$\Lambda-F$$-crystals on $$U/K$$ overconvergent around $$D$$.

### MSC:

 14F30 $$p$$-adic cohomology, crystalline cohomology 14G20 Local ground fields in algebraic geometry 14H25 Arithmetic ground fields for curves

Zbl 0639.14011
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