Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0714.14024
Sekiguchi, T.; Oort, F.; Suwa, N.
On the deformation of Artin-Schreier to Kummer. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 22, No. 3, 345-375 (1989). ISSN 0012-9593

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 $Aut\sb k(C)$ of C. The problem dealt with in this paper is formulated as follows: \par Lift a given pair $(C,G)$ to a pair $({\cal C},G)$ of a smooth proper curve ${\cal C}$ and a subgroup $G\subset Aut({\cal C})$ over a suitable discrete valuation ring A dominating $W(k)$; \par or equivalently, the problem is formulated algebraically:\par Let C/D be a Galois covering of curves over k with Galois group G. Lift C/D to a Galois covering ${\cal C}/{\cal D}$ over a suitable discrete valuation ring A dominating W(k).\par The problem has a positive answer if $C\vert D$ 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: \par 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 $({\cal C},\sigma)$ of $(C,\sigma)$ over $W(k)[\xi]$ where $\xi$ is a primitive p-th root of unity. \par This is proved using class field theory, that is, combining the Artin- Schreier sequence $0\to {\bbfZ}/p{\bbfZ}\to {\bbfG}\sb a\to\sp{p}{\bbfG}\sb a\to 0$ and the Kummer sequence $0\to \mu\sb p\to {\cal G}\sb m\to\sp{p}{\bbfG}\sb m\to 0$.
[N.Yui]

MSC 2000:: *14H30 Coverings, fundamental group (curves)
13K05 Witt vectors and related rings
14E07 Birational automorphisms, Cremona group and generalizations
11S31 Class field theory for local fields

Keywords: lifting problem; deformation of ${\bbfG}\sb a$ to ${\bbfG}\sb m$; characteristic p; automorphism group; Galois covering of curves; class field theory; Artin-Schreier sequence; Kummer sequence

Cited in: Zbl 1233.12011 Zbl 1123.14015 Zbl 0923.14006 Zbl 0920.14023 Zbl 0876.14031 Zbl 0735.14033 Zbl 0755.14014

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]