Analytification and Tropicalization Over Non-archimedean Fields

Werner, Annette

doi:10.1007/978-3-319-30945-3_6

Cited by 5 publications

(5 citation statements)

References 23 publications

(58 reference statements)

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“…In this case, continuity for the section map on tropical Grassmannians was shown directly in [CHW14]. See also [Wer16] for an expository account of this construction. Note that in [CHW14], we use log instead of − log for tropicalization.…”

Section: Existence Of the Sectionmentioning

confidence: 94%

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Tropical Skeletons

Gubler¹,

Rabinoff²,

Werner³

2017

Ann. inst. Fourier

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In this paper, we study the interplay between tropical and analytic geometry for closed subschemes of toric varieties. Let K be a complete non-Archimedean field, and let X be a closed subscheme of a toric variety over K. We define the tropical skeleton of X as the subset of the associated Berkovich space X an which collects all Shilov boundary points in the fibers of the Kajiwara-Payne tropicalization map. We develop polyhedral criteria for limit points to belong to the tropical skeleton, and for the tropical skeleton to be closed. We apply the limit point criteria to the question of continuity of the canonical section of the tropicalization map on the multiplicity-one locus. This map is known to be continuous on all torus orbits; we prove criteria for continuity when crossing torus orbits. When X is schön and defined over a discretely valued field, we show that the tropical skeleton coincides with a skeleton of a strictly semistable pair, and is naturally isomorphic to the parameterizing complex of Helm-Katz.

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Section: Existence Of the Sectionmentioning

confidence: 94%

“…This section is defined by associating to every point ω of tropical multiplicity one the unique Shilov boundary point in the fiber of tropicalization over ω. For an overview of these results, see [Wer16].…”

Section: Introductionmentioning

confidence: 99%

Tropical Skeletons

Gubler¹,

Rabinoff²,

Werner³

2017

Ann. inst. Fourier

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“…• Underlying structure of Berkovich spaces: [7], [8], [9, Chapter 1], [18], [19], [25], [54], [55], [56], [77], [79], [131], [132], [168]. See also the survey [172] in this volume. • Potential theory for Berkovich curves: [6, chapter by Baker], [9], [161].…”

Section: Appendix B Thematic Bibliographymentioning

confidence: 99%

Convergence Polygons for Connections on Nonarchimedean Curves

Kedlaya

2016

Simons Symposia

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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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“…For more details and a discussion of higher dimensional skeletons, we refer the reader to the excellent survey by A.Werner[Wer16].…”

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confidence: 99%

Schottky spaces and universal Mumford curves over $${\mathbb {Z}}$$

Poineau

Turchetti

2022

Sel. Math. New Ser.

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For every integer g ≥ 1 we define a universal Mumford curve of genus g in the framework of Berkovich spaces over Z. This is achieved in two steps: first, we build an analytic space Sg that parametrizes marked Schottky groups over all valued fields. We show that Sg is an open, connected analytic space over Z. Then, we prove that the Schottky uniformization of a given curve behaves well with respect to the topology of Sg, both locally and globally. As a result, we can define the universal Mumford curve Cg as a relative curve over Sg such that every Schottky uniformized curve can be described as a fiber of a point in Sg. We prove that the curve Cg is itself uniformized by a universal Schottky group acting on the relative projective line P 1Sg . Finally, we study the action of the group Out(Fg) of outer automorphisms of the free group with g generators on Sg, describing the quotient Out(Fg)\Sg in the archimedean and non-archimedean cases. We apply this result to compare the non-archimedean Schottky space with constructions arising from geometric group theory and the theory of moduli spaces of tropical curves.

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Analytification and Tropicalization Over Non-archimedean Fields

Cited by 5 publications

References 23 publications

Tropical Skeletons

Tropical Skeletons

Convergence Polygons for Connections on Nonarchimedean Curves

Schottky spaces and universal Mumford curves over $${\mathbb {Z}}$$

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