Order \(p\) automorphisms of the open disc of a \(p\)-adic field. (English) Zbl 0923.14007

This paper is in some sense a supplement to an earlier paper [B. Green and M. Matignon, Compos. Math. 113, No. 3, 237-272 (1998; see the preceding review)] of the same authors. It contains a systematic study of automorphisms of order \(p\) of the formal power series ring \(R[[Z]]\) in the following situation: \(k\) is an algebraically closed field of characteristic \(p>0\) and \(R\) a complete discrete valuation ring dominating the ring \(W(k)\) of Witt vectors of \(k\). As the title suggests \(D^0:=\text{Spec} R[[Z]]\) is geometrically interpreted as open unit disk.
With an automorphism \(\sigma\) as above the authors associate the number \(m+1\) of fixed points of \(\sigma\) (in an algebraic closure of \(\text{Frac}(k))\) and the so-called “Hurwitz data” \(h_i \in(\mathbb{Z}/p\mathbb{Z})\), \(i=0,\dots,m\): if \(\zeta\) is a fixed primitive \(p\)-th root of unity (assumed to be contained in \(R)\), and in \(Z_i\) is the \(i\)-th fixed point of \(\sigma\), then \(h_i\) is defined as the exponent of \(\zeta\) by which \(\sigma\) acts on the tangent space at \(Z_i\). If the number of fixed points is small compared to the residue characteristic (i.e. \(0<m<p)\) the authors obtain explicit lists of the possible Hurwitz data. Moreover they show that in this case there are only finitely many conjugacy classes of automorphisms with a given Hurwitz datum.
The authors also consider the minimal semistable model of \(D^0_K\) for which all fixed points of \(\sigma\) specialize to different smooth points. Its reduction is a tree of projective lines which for \(0<m<p\) is shown to be reduced to a single component. For larger \(m\) the paper contains several examples of more complicated trees as reduction.
The authors also give an application to the problem of lifting automorphism groups from \(k\) to \(R\).


14H37 Automorphisms of curves
14G20 Local ground fields in algebraic geometry
14E07 Birational automorphisms, Cremona group and generalizations
13F25 Formal power series rings
14L05 Formal groups, \(p\)-divisible groups
14E22 Ramification problems in algebraic geometry


Zbl 0923.14006
Full Text: DOI


[1] 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). · Zbl 0547.13001
[2] 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.
[3] Robert Coleman and William McCallum, Stable reduction of Fermat curves and Jacobi sum Hecke characters, J. Reine Angew. Math. 385 (1988), 41 – 101. · Zbl 0654.12003
[4] Richard M. Crew, Etale \?-covers in characteristic \?, Compositio Math. 52 (1984), no. 1, 31 – 45. · Zbl 0558.14009
[5] M. Deuring, Automorphismen und Divisorenklassen der Ordnung \(\ell \) in algebraischen Funktionenkörpern, Math. Ann. 113 (1936), 208-215.
[6] M. Deuring, Invarianten und Normalformen elliptischer Funktionenkörper, Math. Zeit. 47 (1941), 47-56.
[7] M. Garuti, Prolongement de revêtement galoisiens en géométrie rigide, Compositio Math. 104 (1996), 305-331. CMP 97:05
[8] B. Green, M. Matignon, Liftings of Galois Covers of Smooth Curves, Compositio Math. 113 (1998), 239-274. · Zbl 0923.14006
[9] Michiel Hazewinkel, Formal groups and applications, Pure and Applied Mathematics, vol. 78, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. · Zbl 0454.14020
[10] M. Matignon, \(p\)-groupes abéliens de type \((p,...,p)\) et disques ouverts \(p\)-adiques, Prépublication 83 (1998), Laboratoire de Mathématiques pures de Bordeaux. · Zbl 0953.12004
[11] James S. Milne, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 1980. · Zbl 0433.14012
[12] 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. · doi:10.1090/pspum/046.2/927980
[13] 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. · Zbl 0714.14024
[14] M. Raynaud, Revêtements de la droite affine en caractéristique \?&gt;0 et conjecture d’Abhyankar, Invent. Math. 116 (1994), no. 1-3, 425 – 462 (French). · Zbl 0798.14013 · doi:10.1007/BF01231568
[15] Michel Raynaud, Mauvaise réduction des courbes et \?-rang, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 12, 1279 – 1282 (French, with English and French summaries). · Zbl 0834.14014
[16] Michel Raynaud, \?-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). · Zbl 0722.14013 · doi:10.1007/978-0-8176-4576-2_7
[17] M. Raynaud, Letter to the authors, November 15, 1996.
[18] Peter Roquette, Abschätzung der Automorphismenanzahl von Funktionenkörpern bei Primzahlcharakteristik, Math. Z. 117 (1970), 157 – 163 (German). · Zbl 0194.35302 · doi:10.1007/BF01109838
[19] I.R. Šafarevičh, On \(p\)-extensions, AMS Transl. series II, 4 (1954), 59-72.
[20] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. 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. · Zbl 0585.14026
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