## 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.

### MSC:

 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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### References:

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