The local lifting problem for actions of finite groups on curves

Ted Chinburg; Robert Guralnick; David Harbater

Annales scientifiques de l'École Normale Supérieure (2011)

  • Volume: 44, Issue: 4, page 537-605
  • ISSN: 0012-9593

Abstract

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Let k be an algebraically closed field of characteristic p > 0 . We study obstructions to lifting to characteristic 0 the faithful continuous action φ of a finite group G on k [ [ t ] ] . To each such  φ a theorem of Katz and Gabber associates an action of G on a smooth projective curve Y over k . We say that the KGB obstruction of φ vanishes if G acts on a smooth projective curve X in characteristic  0 in such a way that X / H and Y / H have the same genus for all subgroups H ⊂ G . We determine for which G the KGB obstruction of every φ vanishes. We also consider analogous problems in which one requires only that an obstruction to lifting φ due to Bertin vanishes for some φ , or for all sufficiently ramified φ . These results provide evidence for the strengthening of Oort’s lifting conjecture which is discussed in [8, Conj. 1.2].

How to cite

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Chinburg, Ted, Guralnick, Robert, and Harbater, David. "The local lifting problem for actions of finite groups on curves." Annales scientifiques de l'École Normale Supérieure 44.4 (2011): 537-605. <http://eudml.org/doc/272151>.

@article{Chinburg2011,
abstract = {Let $k$ be an algebraically closed field of characteristic $p &gt; 0$. We study obstructions to lifting to characteristic $0$ the faithful continuous action $\phi $ of a finite group $G$ on $k[[t]]$. To each such $\phi $ a theorem of Katz and Gabber associates an action of $G$ on a smooth projective curve $Y$ over $k$. We say that the KGB obstruction of $\phi $ vanishes if $G$ acts on a smooth projective curve $X$ in characteristic $0$ in such a way that $X/H$ and $Y/H$ have the same genus for all subgroups $H \subset G$. We determine for which $G$ the KGB obstruction of every $\phi $ vanishes. We also consider analogous problems in which one requires only that an obstruction to lifting $\phi $ due to Bertin vanishes for some $\phi $, or for all sufficiently ramified $\phi $. These results provide evidence for the strengthening of Oort’s lifting conjecture which is discussed in [8, Conj. 1.2].},
author = {Chinburg, Ted, Guralnick, Robert, Harbater, David},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {Galois groups; curves; automorphisms; characteristic $p$; lifting; Oort conjecture},
language = {eng},
number = {4},
pages = {537-605},
publisher = {Société mathématique de France},
title = {The local lifting problem for actions of finite groups on curves},
url = {http://eudml.org/doc/272151},
volume = {44},
year = {2011},
}

TY - JOUR
AU - Chinburg, Ted
AU - Guralnick, Robert
AU - Harbater, David
TI - The local lifting problem for actions of finite groups on curves
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2011
PB - Société mathématique de France
VL - 44
IS - 4
SP - 537
EP - 605
AB - Let $k$ be an algebraically closed field of characteristic $p &gt; 0$. We study obstructions to lifting to characteristic $0$ the faithful continuous action $\phi $ of a finite group $G$ on $k[[t]]$. To each such $\phi $ a theorem of Katz and Gabber associates an action of $G$ on a smooth projective curve $Y$ over $k$. We say that the KGB obstruction of $\phi $ vanishes if $G$ acts on a smooth projective curve $X$ in characteristic $0$ in such a way that $X/H$ and $Y/H$ have the same genus for all subgroups $H \subset G$. We determine for which $G$ the KGB obstruction of every $\phi $ vanishes. We also consider analogous problems in which one requires only that an obstruction to lifting $\phi $ due to Bertin vanishes for some $\phi $, or for all sufficiently ramified $\phi $. These results provide evidence for the strengthening of Oort’s lifting conjecture which is discussed in [8, Conj. 1.2].
LA - eng
KW - Galois groups; curves; automorphisms; characteristic $p$; lifting; Oort conjecture
UR - http://eudml.org/doc/272151
ER -

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