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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, 29 no. 3 (2017), p. 755-798
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Class. Math.: 11G20, 14F35, 14F30
Keywords: unipotent fundamental groups, function fields, monodromy, $p$-adic cohomology, good reduction

Résumé - Abstract

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 $(\varphi ,\nabla )$-module over the bounded Robba ring $\mathcal{E}_K^\dagger $, whose underlying unipotent group (after base changing to the Amice ring $\mathcal{E}_K$) 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 $\ell $-adic unipotent fundamental group is unramified.


[1] Fabrizio Andreatta, Adrian Iovita & Minhyong Kim, A $p$-adic nonabelian criterion for good reduction of curves, Duke Math. J. 164 (2015), p. 2597-2642 Article
[2] Pierre Berthelot, “Cohomologie rigide et cohomologie rigide à supports propres, première partie”,, 1996
[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. 116 (1999), p. 81-110 Article
[5] Pierre Deligne, Le groupe fondamental de la droite projective moins trois points, in Galois groups over $\mathbb{Q}$ (Berkeley, CA, 1987), Mathematical Sciences Research Institute. Publications, Springer, 1989 Article
[6] Hélène Esnault, Phùng Hô Hai & Xiaotao Sun, On Nori’s fundamental group scheme, Geometry and dynamics of groups and spaces, Progress in Mathematics 265, Birkhäuser, 2008, p. 377–398 Article
[7] Majid Hadian, Motivic fundamental groups and integral points, Duke Math. J. 160 (2011), p. 503-565 Article
[8] Ryoshi Hotta, Kiyoshi Takeuchi & Toshiyuki Tanisaki, $D$-modules, perverse sheaves, and representation theory, Progress in Mathematics 236, Birkhäuser, 2008 Article
[9] Osamu Hyodo & Kazuya Kato, Semi-stable reduction and crystalline cohomology with logarithmic poles, in Périodes $p$-adiques (Bures-sur-Yvette, 1988), Astérisque, Société Mathematique de France, 1994, p. 221-268
[10] Aise Johan de Jong, Homomorphisms of Barsotti-Tate groups and crystals in positive characteristic, Invent. Math. 134 (1998), p. 301-333 Article
[11] Fumiharu Kato, Log smooth deformation theory, Tôhoku Math. J. 48 (1996), p. 317-354 Article
[12] Kazuya Kato, Logarithmic structures of Fontaine-Illusie, in Algebraic Analysis, Geometry and Number Theory (Baltimore, 1989), Supplement to the American Journal of Mathematics, Hopkins, 1989, p. 191-224
[13] Kiran Sridhara Kedlaya, Descent theorems for overconvergent $F$-crystals, Ph. D. Thesis, Massachusetts Institute of Technology (USA), 2000 Article
[14] Christopher Lazda, Relative fundamental groups and rational points, Rend. Semin. Mat. Univ. Padova 134 (2015), p. 1-45 Article
[15] Christopher Lazda & Ambrus Pál, Rigid Cohomology over Laurent Series Fields, Algebra and Applications 21, Springer, 2016 Article
[16] Takayuki Oda, A Note on Ramification of the Galois Representation of the Fundamental Group of an Algebraic Curve, II, J. Number Theory 53 (1995), p. 342-355 Article
[17] Atsushi Shiho, Crystalline fundamental groups I: Isocrystals on log crystalline site and log convergent site, J. Math. Sci., Tokyo 7 (2000), p. 509-656
[18] Nicolas Raymond Stalder, Algebraic monodromy groups of $A$-motives, Ph. D. Thesis, ETH Zürich (Switzerland), 2007 Article
[19] Nobuo Tsuzuki, Slope filtration of quasi-unipotent overconvergent F-isocrystals, Ann. Inst. Fourier 48 (1998), p. 379-412