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, Tome 29 (2017) no. 3, pp. 755-798.

Résumé
Abstract

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.

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.

Reçu le : 2016-05-30
Révisé le : 2017-04-24
Accepté le : 2017-05-11
Publié le : 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

Affiliations des auteurs :

Christopher Lazda ¹

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

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     pages = {755--798},
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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, Tome 29 (2017) no. 3, pp. 755-798. doi : 10.5802/jtnb.1000. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1000/

Bibliographie
Cité par

[1] Fabrizio Andreatta; Adrian Iovita; Minhyong Kim A $p$ -adic nonabelian criterion for good reduction of curves, Duke Math. J., Volume 164 (2015) no. 13, pp. 2597-2642 | DOI | Zbl

[2] Pierre Berthelot Cohomologie rigide et cohomologie rigide à supports propres, première partie (1996) (http://perso.univ-rennes1.fr/pierre.berthelot/publis/Cohomologie_Rigide_I.pdf)

[3] Bruno Chiarellotto; Valentina Di Proietto; Atsushi Shiho Comparison of relatively unipotent log de Rham fundamental groups (in preparation)

[4] Bruno Chiarellotto; Bernard Le Stum F-isocristaux unipotents, Compos. Math., Volume 116 (1999) no. 1, pp. 81-110 | DOI | Zbl

[5] Pierre Deligne Le groupe fondamental de la droite projective moins trois points, Galois groups over $ℚ$ (Berkeley, CA, 1987) (Mathematical Sciences Research Institute. Publications), Volume 16 (1989), x+449 pages | DOI | Zbl

[6] Hélène Esnault; Phùng Hô Hai; Xiaotao Sun On Nori’s fundamental group scheme, Geometry and dynamics of groups and spaces (Progress in Mathematics), Volume 265, Birkhäuser, 2008, pp. 377-398 | DOI | Zbl

[7] Majid Hadian Motivic fundamental groups and integral points, Duke Math. J., Volume 160 (2011) no. 3, pp. 503-565 | DOI | Zbl

[8] Ryoshi Hotta; Kiyoshi Takeuchi; Toshiyuki Tanisaki $D$ -modules, perverse sheaves, and representation theory, Progress in Mathematics, 236, Birkhäuser, 2008, x+407 pages | DOI | Zbl

[9] Osamu Hyodo; Kazuya Kato Semi-stable reduction and crystalline cohomology with logarithmic poles, Périodes $p$ -adiques (Bures-sur-Yvette, 1988) (Astérisque), Volume 223 (1994), pp. 221-268 | Zbl

[10] Aise Johan de Jong Homomorphisms of Barsotti-Tate groups and crystals in positive characteristic, Invent. Math., Volume 134 (1998) no. 2, pp. 301-333 erratum in ibid. 138 (1999), no. 1, p. 225 | DOI | Zbl

[11] Fumiharu Kato Log smooth deformation theory, Tôhoku Math. J., Volume 48 (1996) no. 3, pp. 317-354 | DOI | Zbl

[12] Kazuya Kato Logarithmic structures of Fontaine-Illusie, Algebraic Analysis, Geometry and Number Theory (Baltimore, 1989) (Supplement to the American Journal of Mathematics) (1989), pp. 191-224 | Zbl

[13] Kiran Sridhara Kedlaya Descent theorems for overconvergent $F$ -crystals, Massachusetts Institute of Technology (USA) (2000) (Ph. D. Thesis) | DOI

[14] Christopher Lazda Relative fundamental groups and rational points, Rend. Semin. Mat. Univ. Padova, Volume 134 (2015), pp. 1-45 | DOI | Zbl

[15] Christopher Lazda; Ambrus Pál Rigid Cohomology over Laurent Series Fields, Algebra and Applications, 21, Springer, 2016, x+267 pages | DOI | Zbl

[16] Takayuki Oda A Note on Ramification of the Galois Representation of the Fundamental Group of an Algebraic Curve, II, J. Number Theory, Volume 53 (1995) no. 2, pp. 342-355 | DOI | Zbl

[17] Atsushi Shiho Crystalline fundamental groups I: Isocrystals on log crystalline site and log convergent site, J. Math. Sci., Tokyo, Volume 7 (2000) no. 4, pp. 509-656 | Zbl

[18] Nicolas Raymond Stalder Algebraic monodromy groups of $A$ -motives, ETH Zürich (Switzerland) (2007) (Ph. D. Thesis) | DOI

[19] Nobuo Tsuzuki Slope filtration of quasi-unipotent overconvergent F-isocrystals, Ann. Inst. Fourier, Volume 48 (1998) no. 2, pp. 379-412 | DOI | Zbl

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