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 \(D\).


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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