Stable reduction and uniformization of abelian varieties I

Abstract

9 Springer-Verlag 1985 Proposition 1.2. Let X u be distinguished and assume that Xu is geometrically reduced. Then, for any complete field K extending k, the formal analytic variety X~ is distinguished. Its reduction X~ is canonically isomorphic to the I(-extension of the reduction )(u of X u. Proof. Use [B 1, Satz 6.4]. Proposition 1.3. Assume that X is quasi-compact (i.e., that X admits a finite admissible open affinoid covering), and that X is geometrically reduced. Then there exists a finite separable extension k' of k such that X k' is distinguished and has geometrically reduced reduction )~k'. Proof. Use [B2, Lemma 2.7].Now let k-be the completion of the algebraic closure of k. We say that a formal covering ~B of X~is defined over k if there exists a formal covering Il of X such that = 1/~. Likewise we say that X~ is defined over k.Proposition _1.4. Assume that X is quasi-compact, and let ~ be a formal covering of X k. Then X k is defined over a finite separable extension k' of k.

On the structure of non-archimedean analytic curves

“…The (formal) semistable reduction theory of a smooth complete algebraic curve was worked out carefully in [BL85] in the language of rigid analytic spaces and formal analytic varieties (see Remark 4.2(3)); one can view much of this section as a translation of that paper into our language of semistable vertex sets.…”

“…(3) Let X be a semistable formal R-curve. Since X is reduced, X is a formal analytic variety in the sense of [BL85]. In particular, if Spf(A) is a formal affine open subset of X then A is the ring of power-bounded elements of…”

“…The corresponding semistable formal models X, X ′ were constructed above by finding coverings U , U ′ of X an by affinoid domains whose canonical models glue along the canonical models of their intersections. (Such a covering is called a formal covering in [BL85].) It is clear that if U refines U ′ , in the sense that every affinoid in U is contained in an affinoid in U ′ , then we obtain a morphism X → X ′ of semistable formal models.…”

“…Of course this question essentially reduces to the existence of a stable model of X when X = X; using [BL85] we can also treat the case when X is not proper.…”

Ramification of local fields with imperfect residue fields

“…We may assume that g i = h for any (i, j) ∈ P 2 . We denote by Y the right hand side of (3). Let x = (x i ) i∈I ∈ Y (Ω).…”