A generalization of the Oort Conjecture

Obus, Andrew

doi:10.48550/arxiv.1502.07623

Cited by 4 publications

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“…The argument comes from [Pop14] and a conversation with Pop, but was only written in [Pop14] for G cyclic. The papers [Obu15] and [Obu16] use similar arguments, but do not directly cite [Pop14] since they deal with non-cyclic groups. Our statement here is intended to be citeable for general G.…”

Section: [[T ]]) Such That Normalizing R[[t ]] In the Corresponding K...mentioning

confidence: 99%

Lifting of curves with automorphisms

Obus

2017

Preprint

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The lifting problem for curves with automorphisms asks whether we can lift a smooth projective characteristic p curve with a group G of automorphisms to characteristic zero. This was solved by Grothendieck when G acts with prime-to-p stabilizers, and there has been much progress over the last few decades in the wild case. We survey the techniques and obstructions for this lifting problem, aiming at a reader whose background is limited to scheme theory at the level of Hartshorne's book. Throughout, we include numerous examples and clarifying remarks. We also provide a list of open questions.

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Section: [[T ]]) Such That Normalizing R[[t ]] In the Corresponding K...mentioning

confidence: 99%

Lifting of curves with automorphisms

Obus

2017

Preprint

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“…The depth Swan conductor and the different are closely related. For our main result, we only need to understand this relation for Z/p-covers, but we go a bit deeper here to be able to phrase our main result in terms of Swan conductors (Corollary 5.3), as is the context in [OW14] and [Obu15].…”

Section: Comparison Between Depthmentioning

confidence: 99%

“…✷ Part (ii) of the following corollary is a useful result for the local lifting problem. In particular, it plays a key role in proving the main results of [Obu15].…”

Section: Assumptions On Relative Coversmentioning

confidence: 99%

“…What we prove here generalizes a key technical step from [OW14] needed for the iterative technique to work. The generality we work in applies, for instance, to [Obu15], which builds on [OW14] 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 [OW14].…”

Section: Introductionmentioning

confidence: 99%

“…The key idea in this paper is to rephrase the question of whether the inverse image of a disk is a disk in two different ways: one in terms of Cohen-Temkin-Trushin's Berkovich different ( [CTT14]), and one in terms of Kato's depth Swan conductor ([Kat87a]). Indeed, Theorem 5.2 can be interpreted in terms of either the Berkovich different or Swan conductor, as is done in Corollary 5.3 (the Swan conductor version is the form of the theorem applied in [OW14] and [Obu15]). The maximal r for which the inverse image of D[r] is not a disk corresponds to a kink in a piecewise linear function built from the Berkovich different/Swan conductor, hence the title of the paper.…”

Section: Introductionmentioning

confidence: 99%

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Wild ramification kinks

Obus

Wewers

2016

Res Math Sci

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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].

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Convergence Polygons for Connections on Nonarchimedean Curves

Kedlaya

2016

Nonarchimedean and Tropical Geometry

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In classical analysis, one builds the catalog of special functions by repeatedly adjoining solutions of differential equations whose coefficients are previously known functions. Consequently, the properties of special functions depend crucially on the basic properties of ordinary differential equations. This naturally led to the study of formal differential equations, as in the seminal work of Turrittin [167]; this may be viewed retroactively as a theory of differential equations over a trivially valued field. After the introduction of p-adic analysis in the early 20th century, there began to be corresponding interest in solutions of p-adic differential equations; however, aside from some isolated instances (e.g., the proof of the Nagell-Lutz theorem; see Theorem 3.4), a unified theory of p-adic ordinary differential equations did not emerge until the pioneering work of Dwork on the relationship between p-adic special functions and the zeta functions of algebraic varieties over finite fields (e.g., see [57,58]). At that point, serious attention began to be devoted to a serious discrepancy between the p-adic and complex-analytic theories: on an open p-adic disc, a nonsingular differential equation can have a formal solution which does not converge in the entire disc (e.g., the exponential series). One is thus led to quantify the convergence of power series solutions of differential equations involving rational functions over a nonarchimedean field; this was originally done by Dwork in terms of the generic radius of convergence [59]. This and more refined invariants were studied by numerous authors during the half-century following Dwork's initial work, as documented in the author's book [93].At around the time that [93] was published, a new perspective was introduced by Baldassarri [13] (and partly anticipated in prior unpublished work of Baldassarri and Di Vizio [15]) which makes full use of Berkovich's theory of nonarchimedean analytic spaces. Given a differential equation as above, or more generally a connection on a curve over a nonarchimedean field, one can define an invariant called the convergence polygon; this is a function from the underlying Berkovich topological space of the curve into a space of Newton polygons, which measures the convergence of formal horizontal sections and is well-behaved with respect to both the topology and the piecewise linear structure on the Berkovich space. One can translate much of the prior theory of p-adic differential equations into (deceptively) simple statements about the behavior of the convergence polygon; this process was carried out in a series of papers by Poineau and Pulita [141,133,135,136], as supplemented by work of this author [98] and upcoming joint work with Baldassarri [16].

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A generalization of the Oort Conjecture

Cited by 4 publications

References 0 publications

Lifting of curves with automorphisms

Lifting of curves with automorphisms

Wild ramification kinks

Convergence Polygons for Connections on Nonarchimedean Curves

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