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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Order $p$ automorphisms of the open disc of a $p$-adic field
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by Barry Green and Michel Matignon
J. Amer. Math. Soc. 12 (1999), 269-303


Let $k$ be an algebraically closed field of characteristic $p>0,$ $W(k)$ the ring of Witt vectors and $R$ a complete discrete valuation ring dominating $W(k)$ and containing $\zeta ,$ a primitive $p$-th root of unity. Let $\pi$ denote a uniformizing parameter for $R.$ We study order $p$ automorphisms of the formal power series ring $R[[Z]],$ which are defined by a series \begin{equation*}\sigma (Z)=\zeta Z(1+a_{1}Z+\cdots +a_{i}Z^{i}+\cdots )\in R[[Z]].\end{equation*} The set of fixed points of $\sigma$ is denoted by $F_{\sigma }$ and we suppose that they are $K$-rational and that $|F_{\sigma }|=m+1$ for $m\geq 0.$ Let ${\mathcal {D}}^{o}$ be the minimal semi-stable model of the $p$-adic open disc over $R$ in which $F_{\sigma }$ specializes to distinct smooth points. We study the differential data that can be associated to each irreducible component of the special fibre of ${\mathcal {D}}^{o}.$ Using this data we show that if $m<p$, then the fixed points are equidistant, and that there are only a finite number of conjugacy classes of order $p$ automorphisms in $\operatorname {Aut}_{R}(R[[Z]])$ which are not the identity $\operatorname {mod} (\pi ).$
  • N. Bourbaki, Éléments de mathématique. Fascicule XXVIII. Algèbre commutative. Chapitre 3: Graduations, filtrations et topologies. Chapitre 4: Idéaux premiers associés et décomposition primaire, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1293, Hermann, Paris, 1961 (French). MR 0171800
  • Robert F. Coleman, Torsion points on curves, Galois representations and arithmetic algebraic geometry (Kyoto, 1985/Tokyo, 1986) Adv. Stud. Pure Math., vol. 12, North-Holland, Amsterdam, 1987, pp. 235–247. MR 948246, DOI 10.2969/aspm/01210235
  • Robert Coleman and William McCallum, Stable reduction of Fermat curves and Jacobi sum Hecke characters, J. Reine Angew. Math. 385 (1988), 41–101. MR 931215
  • Richard M. Crew, Etale $p$-covers in characteristic $p$, Compositio Math. 52 (1984), no. 1, 31–45. MR 742696
  • M. Deuring, Automorphismen und Divisorenklassen der Ordnung $\ell$ in algebraischen Funktionenkörpern, Math. Ann. 113 (1936), 208–215.
  • Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
  • M. Garuti, Prolongement de revêtement galoisiens en géométrie rigide, Compositio Math. 104 (1996), 305-331.
  • B. Green, M. Matignon, Liftings of Galois Covers of Smooth Curves, Compositio Math. 113 (1998), 239-274.
  • Michiel Hazewinkel, Formal groups and applications, Pure and Applied Mathematics, vol. 78, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506881
  • M. Matignon, $p$-groupes abéliens de type $(p,...,p)$ et disques ouverts $p$-adiques, Prépublication 83 (1998), Laboratoire de Mathématiques pures de Bordeaux.
  • James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
  • Frans Oort, Lifting algebraic curves, abelian varieties, and their endomorphisms to characteristic zero, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 165–195. MR 927980, DOI 10.1090/pspum/046.2/927980
  • T. Sekiguchi, F. Oort, and N. Suwa, On the deformation of Artin-Schreier to Kummer, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 3, 345–375. MR 1011987, DOI 10.24033/asens.1586
  • M. Raynaud, Revêtements de la droite affine en caractéristique $p>0$ et conjecture d’Abhyankar, Invent. Math. 116 (1994), no. 1-3, 425–462 (French). MR 1253200, DOI 10.1007/BF01231568
  • Michel Raynaud, Mauvaise réduction des courbes et $p$-rang, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 12, 1279–1282 (French, with English and French summaries). MR 1310671
  • Michel Raynaud, $p$-groupes et réduction semi-stable des courbes, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 179–197 (French). MR 1106915, DOI 10.1007/978-0-8176-4576-2_{7}
  • M. Raynaud, Letter to the authors, November 15, 1996.
  • Peter Roquette, Abschätzung der Automorphismenanzahl von Funktionenkörpern bei Primzahlcharakteristik, Math. Z. 117 (1970), 157–163 (German). MR 279100, DOI 10.1007/BF01109838
  • I.R. Šafarevičh, On $p$-extensions, AMS Transl. series II, 4 (1954), 59–72.
  • Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
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Bibliographic Information
  • Barry Green
  • Affiliation: Department of Mathematics, University of Stellenbosch, Stellenbosch, 7602, South Africa
  • MR Author ID: 76490
  • Email:
  • Michel Matignon
  • Affiliation: Mathématiques Pures de Bordeaux, UPRS-A 5467, C.N.R.S Université de Bordeaux I, 351, cours de la Libération 33405 – Talence, Cedex, France
  • Email:
  • Received by editor(s): November 25, 1997
  • Received by editor(s) in revised form: June 24, 1998
  • © Copyright 1999 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 12 (1999), 269-303
  • MSC (1991): Primary 14G20, 14L27; Secondary 14D15, 14E22
  • DOI:
  • MathSciNet review: 1630112