Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Order $p$ automorphisms of the open disc
of a $p$-adic field


Authors: Barry Green and Michel Matignon
Journal: J. Amer. Math. Soc. 12 (1999), 269-303
MSC (1991): Primary 14G20, 14L27; Secondary 14D15, 14E22
MathSciNet review: 1630112
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: 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 $\operatorname{mod} (\pi ).$


References [Enhancements On Off] (What's this?)

  • [B] N. Bourbaki, Éléments de mathématique. Fascicule XXVIII. Algèbre commutative. Chapitre 3: Graduations, filtra- tions et topologies. Chapitre 4: Idéaux premiers associés et décomposition primaire, Actualités Scientifiques et Industrielles, No. 1293, Hermann, Paris, 1961 (French). MR 0171800 (30 #2027)
  • [Co] 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 (89d:11050)
  • [Co-Mc] 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 (89h:11026)
  • [Cr] Richard M. Crew, Etale 𝑝-covers in characteristic 𝑝, Compositio Math. 52 (1984), no. 1, 31–45. MR 742696 (85f:14011)
  • [D1] M. Deuring, Automorphismen und Divisorenklassen der Ordnung $\ell $ in algebraischen Funktionenkörpern, Math. Ann. 113 (1936), 208-215.
  • [D2] Max Deuring, Invarianten und Normalformen elliptischer Funktionenkörper, Math. Z. 47 (1940), 47–56 (German). MR 0006172 (3,266d)
  • [Ga] M. Garuti, Prolongement de revêtement galoisiens en géométrie rigide, Compositio Math. 104 (1996), 305-331. CMP 97:05
  • [G-M] B. Green, M. Matignon, Liftings of Galois Covers of Smooth Curves, Compositio Math. 113 (1998), 239-274.
  • [H] Michiel Hazewinkel, Formal groups and applications, Pure and Applied Mathematics, vol. 78, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1978. MR 506881 (82a:14020)
  • [M] M. Matignon, $p$-groupes abéliens de type $(p,...,p)$ et disques ouverts $p$-adiques, Prépublication 83 (1998), Laboratoire de Mathématiques pures de Bordeaux.
  • [Mi] James S. Milne, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531 (81j:14002)
  • [O] 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 (89c:14069)
  • [O-S-S] 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 (91g:14041)
  • [Ra1] 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 (94m:14034), http://dx.doi.org/10.1007/BF01231568
  • [Ra2] 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 (95k:14026)
  • [Ra3] 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 (92m:14025), http://dx.doi.org/10.1007/978-0-8176-4576-2_7
  • [Ra4] M. Raynaud, Letter to the authors, November 15, 1996.
  • [Ro] Peter Roquette, Abschätzung der Automorphismenanzahl von Funktionenkörpern bei Primzahlcharakteristik, Math. Z. 117 (1970), 157–163 (German). MR 0279100 (43 #4826)
  • [Sa] I.R. \v{S}afarevi\v{c}h, On $p$-extensions, AMS Transl. series II, 4 (1954), 59-72.
  • [Si] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210 (87g:11070)
    Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1992. Corrected reprint of the 1986 original. MR 1329092 (95m:11054)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 14G20, 14L27, 14D15, 14E22

Retrieve articles in all journals with MSC (1991): 14G20, 14L27, 14D15, 14E22


Additional Information

Barry Green
Affiliation: Department of Mathematics, University of Stellenbosch, Stellenbosch, 7602, South Africa
Email: bwg@land.sun.ac.za

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: matignon@math.u-bordeaux.fr

DOI: http://dx.doi.org/10.1090/S0894-0347-99-00284-2
PII: S 0894-0347(99)00284-2
Keywords: Order $p$ automorphisms of open $p$-adic discs, fixed points, Hurwitz data, Lubin-Tate formal groups, semi-stable models, degeneration of $\mu _{p}$-torsors, automorphism conjugacy classes
Received by editor(s): November 25, 1997
Received by editor(s) in revised form: June 24, 1998
Article copyright: © Copyright 1999 American Mathematical Society