×

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)

References:

[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 · doi:10.4171/rsmup/128-7
[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 · doi:10.1007/s002080050031
[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 · doi:10.1215/S0012-7094-99-09708-9
[6] P. Colmez and J.-M. Fontaine, Construction des représentations \(p\)-adiques semi-stables , Invent. Math. 140 (2000), 1-43. · Zbl 1010.14004 · doi:10.1007/s002220000042
[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. · doi:10.1007/978-1-4613-9649-9_3
[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 · doi:10.2307/2007012
[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 · doi:10.1215/00127094-1444296
[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 · doi:10.1007/BF02684688
[18] J. Labute, On the descending central series of groups with a single defining relation , J. Algebra 14 (1970), 16-23. · Zbl 0198.34601 · doi:10.1016/0021-8693(70)90130-4
[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 · doi:10.1006/jnth.1995.1095
[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 · doi:10.1090/S0065-9266-2010-00625-2
[21] D. G. Quillen, On the associated graded ring of a group ring , J. Algebra 10 (1968), 411-418. · Zbl 0192.35803 · doi:10.1016/0021-8693(68)90069-0
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.