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

Lepage, Emmanuel

Annales de l'Institut Fourier, Tome 63 (2013) no. 4, pp. 1427-1467.

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 $\left(p^{\prime }\right)$ 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 $\left(p^{\prime }\right)$ 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.

[1] André, Yves On a geometric description of Gal$({\overline{\mathbf{Q}}}_p/{\mathbf{Q}}_p)$ and a $p$-adic avatar of $\widehat{GT}$, Duke Math. J., Volume 119 (2003) no. 1, pp. 1-39 | DOI | MR | Zbl

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

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

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

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

[6] Grothendieck, Alexander Revêtements étales et groupe fondamental (SGA1), Lecture Notes in Mathematics, 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 | Zbl

[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) (Astérisque), Volume 279, Société Mathématique de France, 2002, pp. 271-322 | Numdam | MR | Zbl

[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, Volume 14 (2007) no. 1, pp. 113-116 | MR | Zbl

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

[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 | Zbl

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

[13] Kulikov, Vik. S. Degenerations of $K3$ surfaces and Enriques surfaces, Izv. Akad. Nauk SSSR Ser. Mat., Volume 11 (1977) no. 5, pp. 957-989 | MR | Zbl

[14] Lepage, Emmanuel Coverings in $p$-adic analytic geometry and log coverings II: Cospecialization of the $\left(p^{\prime }\right)$-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., Volume 46 (2010) no. 4, pp. 849-897 | MR | Zbl

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

[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

[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

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

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

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

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