Fundamental groups and good reduction criteria for curves over positive characteristic local fields

Christopher Lazda

doi:10.5802/jtnb.1000

Christopher Lazda ¹

¹ KdVI - Universiteit van Amsterdam P.O. Box 94248 1090 GE Amsterdam, the Netherlands

Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 3, pp. 755-798.

Abstract
Résumé

In this article I define and study the overconvergent rigid fundamental group of a variety over an equicharacteristic local field. This is a non-abelian $(ϕ, \nabla)$ -module over the bounded Robba ring $ℰ_{K}^{†}$ , whose underlying unipotent group (after base changing to the Amice ring $ℰ_{K}$ ) is exactly the classical rigid fundamental group. I then use this to prove an equicharacteristic, $p$ -adic analogue of Oda’s theorem that a semistable curve over a $p$ -adic field has good reduction if and only if the Galois action on its $ℓ$ -adic unipotent fundamental group is unramified.

Dans cette article je défine et étudie le groupe fondamental rigide surconvergent d’une variété algébrique sur un corps local d’équicaractéristique. C’est un $(ϕ, \nabla)$ -module non-abélien sur l’anneu de Robba borné $ℰ_{K}^{†}$ , qui donne le groupe fondamental rigide classique après un changement de base vers l’anneau d’Amice $ℰ_{K}$ . Puis je prouve un ananlogue $p$ -adique d’un théorème d’Oda qui dit qu’une courbe semistable sur un corps $p$ -adique a bonne réduction si et seulement si l’action de Galois sur le groupe fondamental rendu unipotent est non-ramifiée.

Received: 2016-05-30
Revised: 2017-04-24
Accepted: 2017-05-11
Published online: 2017-12-12

DOI: 10.5802/jtnb.1000

Classification: 11G20, 14F35, 14F30
Mots-clés : unipotent fundamental groups, function fields, monodromy, $p$-adic cohomology, good reduction

Author's affiliations:

Christopher Lazda ¹

¹ KdVI - Universiteit van Amsterdam P.O. Box 94248 1090 GE Amsterdam, the Netherlands

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Christopher Lazda. Fundamental groups and good reduction criteria for curves over positive characteristic local fields. Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 3, pp. 755-798. doi : 10.5802/jtnb.1000. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1000/

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