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.

Classification : 11G20, 14H30, 14G22
Keywords: fundamental groups, Berkovich spaces, specialization
Mot clés : groupes fondamentaux, espaces de Berkovich, spécialisation

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