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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 \(R\).

MSC:

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

Citations:

Zbl 0923.14006
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References:

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