Covers in $p$-adic analytic geometry and log covers I: Cospecialization of the $(p^{\prime})$-tempered fundamental group for a family of curves

Lepage, Emmanuel

Covers in

p

-adic analytic geometry and log covers I: Cospecialization of the

(p^{'})

-tempered fundamental group for a family of curves
[Revêtements en géométrie analytique

p

-adique et revêtements logarithmiques : cospécialisation du groupe fondamental

(p^{'})

-tempéré pour une famille de courbes]

Lepage, Emmanuel

Annales de l'Institut Fourier, Tome 63 (2013), p. 1427-1467 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé
Abstract

Le groupe fondamental tempéré d’un espace analytique $p$ -adique classifie les revêtements qui sont dominés par un revêtement topologique (pour la topologie de Berkovich) d’un revêtement étale fini de cet espace. Nous construisons ici des morphismes de cospécialisation entre les versions $(p^{'})$ du groupe fondamental tempéré des fibres d’une famille lisse avec réduction semistable. Pour ce faire, nous traduisons notre problème en termes de morphismes de cospécialisation de groupes fondamentaux des fibres logarithmiques de la réduction modulo $p$ et prouvons l’invariance du groupe fondamental logarithmique géométrique d’un log-schéma log-lisse au-dessus d’un point logarithmique par changement de base.

The tempered fundamental group of a $p$ -adic analytic space classifies covers that are dominated by a topological cover (for the Berkovich topology) of a finite étale cover of the space. Here we construct cospecialization homomorphisms between $(p^{'})$ versions of the tempered fundamental groups of the fibers of a smooth family of curves with semistable reduction. To do so, we will translate our problem in terms of cospecialization morphisms of fundamental groups of the log fibers of the log reduction and we will prove the invariance of the geometric log fundamental group of log smooth log schemes over a log point by change of log point.

Publié le : 2013-01-01

MR 3137359 | Zbl 06359593

DOI : https://doi.org/10.5802/aif.2807
Classification: 11G20, 14H30, 14G22
Mots clés: groupes fondamentaux, espaces de Berkovich, spécialisation

@article{AIF_2013__63_4_1427_0,
     author = {Lepage, Emmanuel},
     title = {Covers in $p$-adic analytic geometry and log covers I: Cospecialization of the $(p^{\prime})$-tempered fundamental group for a family of curves},
     journal = {Annales de l'Institut Fourier},
     volume = {63},
     year = {2013},
     pages = {1427-1467},
     doi = {10.5802/aif.2807},
     zbl = {06359593},
     mrnumber = {3137359},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2013__63_4_1427_0}
}

Lepage, Emmanuel. Covers in $p$-adic analytic geometry and log covers I: Cospecialization of the $(p^{\prime})$-tempered fundamental group for a family of curves. Annales de l'Institut Fourier, Tome 63 (2013) pp. 1427-1467. doi : 10.5802/aif.2807. http://gdmltest.u-ga.fr/item/AIF_2013__63_4_1427_0/

Bibliographie

[1] André, Yves On a geometric description of Gal $({\bar{Q}}_{p} / Q_{p})$ and a $p$ -adic avatar of $\hat{G T}$ , Duke Math. J., Tome 119 (2003) no. 1, pp. 1-39 | Article | MR 1991645 | Zbl 1155.11356

[2] André, Yves Period mappings and differential equations. From $ℂ$ to $ℂ_{p}$ , Mathematical Society of Japan, Tokyo, MSJ Memoirs, Tome 12 (2003) (Tôhoku-Hokkaidô lectures in arithmetic geometry, With appendices by F. Kato and N. Tsuzuki) | MR 1978691

[3] Berkovich, Vladimir G. Spectral theory and analytic geometry over non-Archimedean fields, American Mathematical Society, Providence, RI, Mathematical Surveys and Monographs, Tome 33 (1990) | MR 1070709 | Zbl 0715.14013

[4] Berkovich, Vladimir G. Smooth $p$ -adic analytic spaces are locally contractible, Invent. Math., Tome 137 (1999) no. 1, pp. 1-84 | Article | MR 1702143 | Zbl 0930.32016

[5] Giraud, Jean Cohomologie non abélienne, Springer-Verlag, Grundlehren der mathematischen Wissenschaften, Tome 179 (1971) | MR 344253 | Zbl 0226.14011

[6] Grothendieck, Alexander Revêtements étales et groupe fondamental (SGA1), Springer, Berlin, Lecture Notes in Mathematics, Tome 224 (1971)

[7] Grothendieck, Alexander; Dieudonné, J. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas (Quatrième partie), Inst. Hautes Études Sci., Publications Mathématiques (1967) no. 31, pp. 5-361 | Numdam | MR 238860 | Zbl 0153.22301

[8] Illusie, Luc An Overview of the works of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology, Cohomologies $p$ -adiques et applications arithmétiques (II), Société Mathématique de France (Astérisque) Tome 279 (2002), pp. 271-322 | MR 1922832 | Zbl 1052.14005

[9] Illusie, Luc; Kato, Kazuya; Nakayama, Chikara Erratum to: Quasi-unipotent logarithmic Riemann-Hilbert correspondences [J. Math. Sci. Univ. Tokyo 12 (2005), no. 1, 1–66; MR2126784], J. Math. Sci. Univ. Tokyo, Tome 14 (2007) no. 1, pp. 113-116 | MR 2320387 | Zbl 1082.14024

[10] De Jong, A. J. Étale fundamental groups of non-Archimedean analytic spaces, Compositio Math., Tome 97 (1995) no. 1-2, pp. 89-118 (Special issue in honour of Frans Oort) | Numdam | MR 1355119 | Zbl 0864.14012

[11] Kato, Kazuya Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD (1989), pp. 191-224 | MR 1463703 | Zbl 0776.14004

[12] Kisin, Mark Prime to $p$ fundamental groups, and tame Galois actions, Ann. Inst. Fourier (Grenoble), Tome 50 (2000) no. 4, pp. 1099-1126 | Article | Numdam | MR 1799739 | Zbl 0961.14014

[13] Kulikov, Vik. S. Degenerations of $K 3$ surfaces and Enriques surfaces, Izv. Akad. Nauk SSSR Ser. Mat., Tome 11 (1977) no. 5, pp. 957-989 | MR 506296 | Zbl 0387.14007

[14] Lepage, Emmanuel Coverings in $p$ -adic analytic geometry and log coverings II: Cospecialization of the $(p^{'})$ -tempered fundamental group in higher dimensions (http://arxiv.org/abs/0903.2349)

[15] Lepage, Emmanuel Tempered fundamental group and metric graph of a Mumford curve, Publ. Res. Inst. Math. Sci., Tome 46 (2010) no. 4, pp. 849-897 | MR 2791009 | Zbl 1213.14047

[16] Mochizuki, Shinichi Semi-graphs of anabelioids, Publ. Res. Inst. Math. Sci., Tome 42 (2006) no. 1, pp. 221-322 | Article | MR 2215441 | Zbl 1113.14025

[17] Ogus, Arthur Lectures on logarithmic algebraic geometry http://math.berkeley.edu/~ogus/preprints/log_book/logbook.pdf (Notes préliminaires, http://math.berkeley.edu/~ogus/preprints/log_book/logbook.pdf)

[18] Olsson, Martin Ch. Log algebraic stacks and moduli of log schemes, ProQuest LLC, Ann Arbor, MI (2001) (Thesis (Ph.D.)–University of California, Berkeley) | MR 2702292

[19] Orgogozo, Fabrice Erratum et compléments à l’article Altérations et groupe fondamental premier à $p$ paru au Bulletin de la S.M.F. (131), tome 1, 2003, unpublished (http://hal.archives-ouvertes.fr/docs/00/19/66/31/PDF/Alterations_et_groupe_fondamental_premier_a_p_erratum_et_complements_Orgogozo.pdf) | Numdam | MR 1975807

[20] Orgogozo, Fabrice Altérations et groupe fondamental premier à $p$ , Bull. Soc. Math. France, Tome 131 (2003) no. 1, pp. 123-147 | Numdam | MR 1975807 | Zbl 1083.14506

[21] Raynaud, Michel Anneaux locaux henséliens, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Vol. 169 (1970) | MR 277519 | Zbl 0203.05102

[22] Stix, Jakob Projective anabelian curves in positive characteristic and descent theory for log-étale covers, Universität Bonn Mathematisches Institut, Bonn, Bonner Mathematische Schriften [Bonn Mathematical Publications], 354 (2002) (Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 2002) | MR 2012864 | Zbl 1077.14040

[23] Tsuji, Takeshi Saturated morphisms of logarithmic schemes (1997) (unpublished)