A \(p\)-adic nonabelian criterion for good reduction of curves. (English) Zbl 1347.11051

Given a proper smooth curve \(X\) over a complete discrete valuation field \(K\) with characteristic zero. Suppose \(X\) has a semistable model over the valuation ring of \(K\). In this paper, the authors discuss these three categories over \(X\) in a unified manner, the Kummer étale category, the de Rham category, and the crystalline category. For the Tannakian categories, they study the corresponding fundamental groups, particularly, the unipotent \(p\) -adic étale fundamental group \(G^{\text{ét}}\) and the unipotent de Rham fundamental group \(G^{\mathrm{dR}}\) of \(X\), respectively.
Here in the paper, \(G^{\text{ét}}\) is said to be a crystalline fundamental group if \(G^{\text{ét}}\) and \(G^{\mathrm{dR}}\) respectively tensor the Fontaine’s rings \( B_{\mathrm{crys},K}\) and \(B_{\mathrm{st},K}\) are \(G\left( \bar{K}/K\right) \)-equivariant isomorphic group schemes. Then, the authors obtain the main theorem of the paper, i.e., a criterion for good reduction of curves: The curve \(X\) has good reduction if and only if the fundamental group \(G^{\text{ét}}\) is crystalline. This result is a generalization of many known results on good reduction.


11G20 Curves over finite and local fields
14F30 \(p\)-adic cohomology, crystalline cohomology
14G22 Rigid analytic geometry
14G32 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)
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