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p-groupes et réduction semi-stable des courbes. (p-groups and semistable reduction of curves). (French) Zbl 0722.14013

The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. III, Prog. Math. 88, 179-197 (1990).
[For the entire collection see Zbl 0717.00010.]
We briefly summarize the introduction of the paper under review: Let R be a complete discrete valuation ring with residue field k of characteristic \(p>0.\) Let X be a smooth proper curve over R with geometrically connected fibres and let \(Y_ K\) be a finite, étale Galois covering of the generic fibre \(X_ K\) with Galois group G. Let G have order prime to \(p.\) Then, by the theory of the Grothendieck fundamental group, after possibly passing to a tamely ramified extension of R, the curve \(Y_ K\) extends to a covering \(Y\to X\). - Now let G be a p-group. The author then shows:
Theorem 1: 1. The Jacobian of \(Y_ K\) has potentially good reduction.
2. After possibly passing to a finite extension of R, the curve \(Y_ K\) admits a semi-stable reduction. The connected components of the graph of the special fibre are trees.
The author also gives some properties of the crystalline slopes of the special fibre.
Reviewer: D.Goss (Columbus)

MSC:

14H30 Coverings of curves, fundamental group
14F30 \(p\)-adic cohomology, crystalline cohomology

Citations:

Zbl 0717.00010