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On the deformation of Artin-Schreier to Kummer. (English) Zbl 0714.14024
Let k be an algebraically closed field of characteristic \(p>0,\) and let W(k) denote the ring of Witt vectors of k. Let C be a smooth complete algebraic curve of genus \(g\) over k. Let G be a subgroup of the automorphism group \(Aut_ k(C)\) of C. The problem dealt with in this paper is formulated as follows:
Lift a given pair \((C,G)\) to a pair \(({\mathcal C},G)\) of a smooth proper curve \({\mathcal C}\) and a subgroup \(G\subset Aut({\mathcal C})\) over a suitable discrete valuation ring A dominating \(W(k)\);
or equivalently, the problem is formulated algebraically:
Let C/D be a Galois covering of curves over k with Galois group G. Lift C/D to a Galois covering \({\mathcal C}/{\mathcal D}\) over a suitable discrete valuation ring A dominating W(k).
The problem has a positive answer if \(C| D\) is unramified, or tamely ramified. However, if C/D is wildly ramified, the answer is in general negative. The main result of this paper is to prove the following theorem:
Let C be a smooth complete algebraic curve over k and let \(G=<\sigma>\) where \(\sigma\) is an automorphism of C of order pm whith \((p,m)=1\). Then there exists a lifting \(({\mathcal C},\sigma)\) of \((C,\sigma)\) over \(W(k)[\xi]\) where \(\xi\) is a primitive p-th root of unity.
This is proved using class field theory, that is, combining the Artin- Schreier sequence \(0\to {\mathbb{Z}}/p{\mathbb{Z}}\to {\mathbb{G}}_ a\to^{p}{\mathbb{G}}_ a\to 0\) and the Kummer sequence \(0\to \mu_ p\to {\mathcal G}_ m\to^{p}{\mathbb{G}}_ m\to 0\).
Reviewer: N.Yui

MSC:
14H30 Coverings of curves, fundamental group
13K05 Witt vectors and related rings (MSC2000)
14E07 Birational automorphisms, Cremona group and generalizations
11S31 Class field theory; \(p\)-adic formal groups
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References:
[1] N. BOURBAKI , Algèbre commutative , Eléments de Math., 27, 28, 30, 31, Hermann, Paris, 1961 - 1965 .
[2] L. BREEN , Extensions of Abelian Sheaves and Eilenberg-Maclane algebras , Invent. Math., Vol. 9, 1969 , pp. 15-44. MR 41 #3488 | Zbl 0181.26401 · Zbl 0181.26401
[3] L. BREEN , Un théorème d’annulation pour certains des Exti de faisceaux abéliens , Ann. scient. Éc. Norm. Sup., Vol. 8, 1975 , pp. 339-352. Numdam | MR 53 #5595 | Zbl 0313.14001 · Zbl 0313.14001
[4] M. DEMAZURE and P. GABRIEL , Groupes algébriques , Tome 1. Masson, Paris, North-Holland Pub. Comp., Amsterdam, 1970 . Zbl 0203.23401 · Zbl 0203.23401
[5] FGA A. GROTHENDIECK , Fondements de la géométrie algébrique . Séminaire Bourbaki, 1952 - 1962 , Secrétariat Math., Paris, 1962 . Zbl 0239.14002 · Zbl 0239.14002
[6] GB A. GROTHENDIECK , Le groupe de Brauer I, II, III . Dix exposés sur la cohomologie des schémas, North-Holland, Masson, 1968 . · Zbl 0198.25901
[7] EGA A. GROTHENDIECK and J. DIEUDONNÉ , Éléments de géométrie algébrique , Pub. Math. I.H.E.S., 1960 - 1967 . Numdam · Zbl 0203.23301
[8] SGA A. GROTHENDIECK et al., Séminaire de géométrie algébrique , Lecture Notes in Math., Nos. 224, 269, 270, 305, Berlin-Heidelberg-New York, Springer-Verlag, 1971 - 1973 .
[9] H. HASSE , Theorie der relativ-zyklischen algebraischen Funktionenkörper, insbesondere bei endlichen Konstantenkörper . Jour. reine angew. Math. (Crelle), 172, 1935 , pp. 37-54. Article | Zbl 0010.00501 | JFM 60.0097.01 · Zbl 0010.00501
[10] J.-I. IGUSA , Arithmetic Variety of Moduli for Genus Two , Ann. Math., 72, 1960 , pp. 612-649. MR 22 #5637 | Zbl 0122.39002 · Zbl 0122.39002
[11] T. KAMBAYASHI and M. MIYANISHI , On Flat Fibrations by the Affine Line , Illinois J. Math., No. 4, 22, 1978 , pp. 662-671. Article | MR 80f:14028 | Zbl 0406.14012 · Zbl 0406.14012
[12] O. A. LAUDAL and K. LØNSTED , Deformations of Curves I. Moduli for Hyperelliptic Curves. Algebraic Geometry , Proceedings Tromsø, Norway 1977 , Lecture Notes in Math., No. 687, Berlin-Heidelberg-New York, Springer-Verlag, 1978 , pp. 150-167. Zbl 0422.14014 · Zbl 0422.14014
[13] D. MUMFORD , Geometric Invariant Theory. Ergebnisse , Berlin-Heidelberg-New York, Springer-Verlag, 1965 . MR 35 #5451 | Zbl 0147.39304 · Zbl 0147.39304
[14] D. MUMFORD , Abelian Varieties , Tata Inst. Studies in Math., Oxford University Press, 1970 . MR 44 #219 | Zbl 0223.14022 · Zbl 0223.14022
[15] M. NAGATA , Local rings , Interscience Tracts in Pure & Applied Math., 131, J. Wiley, New York, 1962 . MR 27 #5790 | Zbl 0123.03402 · Zbl 0123.03402
[16] S. NAKAJIMA , Action of an Automorphism of Order p on Cohomology Groups of an Algebraic Curves , J. Pure and Applied Alg. 42, 1986 , pp. 85-94. MR 88d:14018 | Zbl 0607.14022 · Zbl 0607.14022
[17] F. OORT , Commutative Group Schemes , Lecture Notes in Math., No. 15, Berlin-Heidelberg-New York, Springer-Verlag, 1966 . MR 35 #4229 | Zbl 0216.05603 · Zbl 0216.05603
[18] F. OORT , Finite Group Schemes, Local Moduli for Abelian Varieties and Lifting Problems , Compos. Math., Vol. 23, 1971 , pp. 265-296 (Also : Algebraic Geometry, Oslo, 1970 , F. Oort ed., Wolters-Noordhoff, 1972 , pp. 223-254). Numdam | MR 46 #186 | Zbl 0239.14018 · Zbl 0239.14018
[19] F. OORT , Singularities of coarse moduli schemes , Sém. Dubreil, 29, 1975 / 1976 ; Lecture Notes in Math., Vol. 586, pp. 61-76, Berlin-Heidelberg-New York, Springer-Verlag, 1977 . Zbl 0349.14003 · Zbl 0349.14003
[20] F. OORT and D. MUMFORD , Deformations and Liftings of Finite Commutative Group Schemes , Invent. Math., Vol. 5, 1968 , pp. 317-334. MR 37 #4085 | Zbl 0179.49901 · Zbl 0179.49901
[21] F. OORT and T. SEKIGUCHI , The canonical lifting of an ordinary Jacobian variety need not a Jacobian variety , J. Math. Soc. Japan, Vol. 38, 1986 , pp. 427-437. Article | MR 87g:14047 | Zbl 0605.14031 · Zbl 0605.14031
[22] M. RAYNAUD , Spécialization du foncteur de Picard , I.H.E.S., Vol. 38, 1970 , pp. 27-76. Numdam | MR 44 #227 | Zbl 0207.51602 · Zbl 0207.51602
[23] P. ROQUETTE , Abschötzung der Automorphismenzahl von Funktionenkörpern bei Primzahlcharakteristik , Math. Z., Vol. 117, 1970 , pp. 157-163. Article | MR 43 #4826 | Zbl 0194.35302 · Zbl 0194.35302
[24] T. SEKIGUCHI and F. OORT , On the deformations of Witt groups to tori. In Algebraic and Topological Theories (to the memory of Dr. T. Miyarta), Kinokuniya Co., Ltd., 1985 , pp. 283-298. Zbl 0800.14023 · Zbl 0800.14023
[25] J.-P. SERRE , Groupes algébriques et corps de classes , Hermann, Paris, 1959 . MR 21 #1973 | Zbl 0097.35604 · Zbl 0097.35604
[26] B. SINGH , On the group of automorphisms of a function field of genus at least two , Jour. of Pure and Applied algebra, Vol. 4, 1974 , pp. 205-229. MR 50 #13047 | Zbl 0284.12007 · Zbl 0284.12007
[27] H. STICHTENOTH , Uber die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharakteristik . Teil I : Eine Abschötzung der Ordnung der Automorphismengruppe. Teil II : Ein spezieller typ von Funktionenkörpern, Arch. Math. Vol. 24, 1973 , pp. 527-544 ; Vol. 25, 1973 , pp. 615-631. MR 53 #8068 | Zbl 0282.14007 · Zbl 0282.14007
[28] B. WEISFEILER , On a case of extensions of group schemes , Trans. Math. Soc., No. 1, 248, 1979 , pp. 171-189. MR 81c:14027 | Zbl 0412.14022 · Zbl 0412.14022
[29] W. WATERHOUSE and B. WEISFEILER , One-dimensional affine group schemes , J. of Alg., Vol. 66, 1980 , pp. 550-568. MR 82e:14057 | Zbl 0452.14013 · Zbl 0452.14013
[30] W. WATERHOUSE , A unified Kummer-Artin-Schreier sequence , Math. Ann., Vol. 277, 1987 , pp. 447-451. MR 88j:14058 | Zbl 0608.12026 · Zbl 0608.12026
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