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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[1] J. Amorós, M. Burger, K. Corlette, D. Kotschick, and D. Toledo, Fundamental Groups of Compact Kähler Manifolds , Math. Surveys Monogr. 44 , Amer. Math. Soc., Providence, 1996. · Zbl 0849.32006
[2] F. Andreatta and A. Iovita, Semistable sheaves and comparison isomorphisms in the semistable case , Rend. Sem. Mat. Univ. Padova 128 (2012), 131-285. · Zbl 1275.14012
[3] P. Berthelot, Cohomologie cristalline des schémas de caractéristique \(p>0\) , Lecture Notes in Math. 407 , Springer, Berlin, 1974. · Zbl 0298.14012
[4] C. Breuil, Représentations \(p\)-adiques semi-stables et transversalité de Griffiths , Math. Ann. 307 (1997), 191-224. · Zbl 0883.11049
[5] R. Coleman and A. Iovita, The Frobenius and monodromy operators for curves and abelian varieties , Duke Math. J. 97 (1999), 171-215. · Zbl 0962.14030
[6] P. Colmez and J.-M. Fontaine, Construction des représentations \(p\)-adiques semi-stables , Invent. Math. 140 (2000), 1-43. · Zbl 1010.14004
[7] P. Deligne, Équations différentielles à points singuliers réguliers , Lecture Notes in Math. 163 , Springer, Berlin, 1970.
[8] P. Deligne, “Le groupe fondamental de la droite projective moins trois points” in Galois Groups over \(\mathbb{Q}\) (Berkeley, 1987) , Math. Sci. Res. Inst. Publ. 16 , Springer, New York, 1989, 79-297.
[9] J.-M. Fontaine, Sur certains types de représentations \(p\)-adiques du group de Galois d’un corps local; construction d’un anneau de Barsotti-Tate , Ann. of Math. (2) 115 (1982), 529-577. · Zbl 0544.14016
[10] J.-M. Fontaine, “Le corps des périodes \(p\)-adiques,” with an appendix by P. Colmez, in Périodes \(p\)-adiques (Bures-sur-Yvette, 1988) , Astérisque 223 , Soc. Math. France, Paris, 1994, 59-111.
[11] A. Grothendieck, Eléments de géométrie algébrique, III: Étude cohomologique des faisceaux cohérents, I , Publ. Math. Inst. Hautes Études Sci. 11 (1961).
[12] M. Hadian, Motivic fundamental groups and integral points , Duke Math. J. 160 (2011), 503-565. · Zbl 1234.14020
[13] O. Hyodo and K. Kato, “Semi-stable reduction and crystalline cohomology with logarithmic poles” in Périodes \(p\)-adiques (Bures-sur-Yvette, 1988) , Astérisque 223 , Soc. Math. France, Paris, 1994, 321-347.
[14] L. Illusie, “An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology” in Cohomologies \(p\)-adiques et applications arithmétiques, II , Astérisque 279 , Soc. Math. France, Paris, 2002, 271-322.
[15] K. Kato, “Logarithmic structures of Fontaine-Illusie” in Algebraic Analysis, Geometry, and Number Theory (Baltimore, 1988) , Johns Hopkins Univ. Press, Baltimore, 1989, 191-224. · Zbl 0776.14004
[16] K. Kato, “Semi-stable reduction and \(p\)-adic étale cohomology” in Périodes \(p\)-adiques (Bures-sur-Yvette, 1988) , Astérisque 223 , Soc. Math. France, Paris, 1994, 269-293.
[17] N. M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin , Publ. Math. Inst. Hautes Études Sci. 39 (1970), 175-232. · Zbl 0221.14007
[18] J. Labute, On the descending central series of groups with a single defining relation , J. Algebra 14 (1970), 16-23. · Zbl 0198.34601
[19] T. Oda, A note on ramification of the Galois representation of the fundamental group of an algebraic curve, II , J. Number Theory 53 (1995), 342-355. · Zbl 0844.14013
[20] M. C. Olsson, Towards non-abelian \(p\)-adic Hodge theory in the good reduction case , Mem. Amer. Math. Soc. 210 (2011), no. 990. · Zbl 1213.14002
[21] D. G. Quillen, On the associated graded ring of a group ring , J. Algebra 10 (1968), 411-418. · Zbl 0192.35803
[22] J. L.-Verdier, Des catégories dérivées des catégories abéliennes , Astérisque 239 , Soc. Math. France, Paris, 1996.
[23] V. Vologodsky, Hodge structures on the fundamental group and its applications to \(p\)-adic integration , Mosc. Math. J. 3 (2003), 205-247. · Zbl 1050.14013
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