Given a branched cover f : Y → X between smooth projective curves over a non-archimedean mixed-characteristic local field and an open rigid disk D ⊂ X , we study the question under which conditions the inverse image f −1 (D) is again an open disk. More generally, if the cover f varies in an analytic family, is this true at least for some member of the family? Our main result gives a criterion for this to happen.Keywords: p-adic disk, Berkovich different, Swan conductor, Rigid-analytic Galois cover Mathematics Subject Classification: 14H30, 14H37, 14D99, 14G22, 11G20
BackgroundThis paper is about inverse images of non-archimedean disks under finite morphismsspecifically, when are they themselves disks?Let X be a smooth, projective curve over a mixed-characteristic (0, p) non-archimedean field K . Our main result (Theorem 5.2) applies to flat families F : Y → X × A of Galois branched covers of X parameterized by a quasi-compact, quasi-separated, rigid-analytic space A (e.g., an affinoid). Let D be an open disk in X (say, given a choice of origin and a metric making it a unit disk), and for r > 0, let D[r] ⊆ D be the closed disk centered at the origin of radius p −r (i.e., the set of all points with valuation at least r). The theorem says that, under mild assumptions about branch loci and connectedness (see the beginning of Sect. 5), if there exists a sequence r 1 , r 2 , . . . decreasing to 0 and points a 1 , a 2 , . .is an open disk. In fact, the main result is slightly more general, allowing X to vary over A in a somewhat prescribed manner (see Assumption 4.10) and allowing F to be a tower of Galois covers.While the above problem is of intrinsic interest as a statement about rigid geometry, we are mainly motivated by the local lifting problem, which asks whether a given action of a finite group G on the germ of a smooth curve in characteristic p lifts to characteristic 0. Our recent paper [19] introduces a new "iterative" technique for solving this problem and solves it when G is cyclic, proving the Oort conjecture. What we prove here generalizes a key technical step from [19] needed for the iterative technique to work. The generality we work in applies, for instance, to [18], which builds on [19] to examine the local lifting problem for metacyclic groups. Indeed, we expect that Theorem 5.2 is sufficiently general to be useful in any solution to the local lifting problem that proceeds via the iterative technique from [19].