Galois covers of the open \(p\)-adic disc (Q2655177)

Let \(R\) be a complete discrete valuation ring of mixed characteristic \((0,p)\), with perfect residue field \(k\) and fraction field \(K\). Let \(Y\to D=\text{Spec} R[[Z]]\) be a regular \(G\)-Galois branched cover of the open unit disc, with \(Y\) normal and with reduced special fiber \(Y_k=Y\times_R k\). The paper provides a characterization of properties of the special fiber \(Y_k\to D_k\) of the cover in terms of the generic fiber \(Y_K\to D_K\) and its specializations. Let \(L/K\) be a Lubin-Tate extension (i.e., \(L=KL_0\) where \(L_0/H\) is a genuine Lubin-Tate extension and \(K/H\) is unramified). Let \(\pi=(\pi_E)_E\) be a prime element of the field of norms \(X_K(L)\), where \(E\) ranges over the finite subextensions of \(L/K\). Choosing \(\pi\) furnishes us with an isomorphism \(k[[z]]\cong R_{X_K(L)}\), where \(R_{X_K(L)}\) is the ring of integers of \(X_K(L)\), and so allows us to view \(Y_k\) as a finite extension of \(R_{X_K(L)}\). Additionally, each \(\pi_E\) is a point of \(D_K\) with residue field \(E\) and so we may consider the fiber \(Y_E\to E\) (there appears to be ambiguity in the author's notation when \(E=K\)). The main theorem ``says that in the abelian case, the irreducibility of the fibers \(Y_E\) implies the irreducibility of the special fiber \(Y_k\). Moreover, for arbitrary groups \(G\), if \(Y_k\) is irreducible, then the separability of the special fiber is determined by the limiting behavior of the differents \(d_E\) of the fibers \(Y_E\to x^E\). Finally, when the special fiber \(Y_k\to D_k\) is separable (but perhaps reducible) its generic fiber \(Y_{k,\eta}\to D_{k,\eta}\) is obtained by applying the field of norms functor to the ``limit'' of the fibers \(Y_E\to x^E\).'' The method of proof is via estimates typical to the theory of the field of norms, such as those used by \textit{J.-P. Wintenberger} [Ann. Sci. Ec. Norm. Super. (4) 16, 59--89 (1983; Zbl 0516.12015)] to prove that the field of norms functor is essentially surjective. In the final section of the paper the author uses his theorem to reformulate a variant of \textit{F. Oort}'s conjecture [Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, part 2, Proc. Symp. Pure Math. 46, 165--195 (1987; Zbl 0645.14017)] concerning the lifting of degree \(m\) cyclic covers of curves in finite characteristic to characteristic zero as a statement about the existence of covers of \(W(\overline{\mathbb{F}_p})[\zeta_m][[Z]]\) satisfying certain arithmetical properties.