A combinatorial Li–Yau inequality and rational points on curves

Cornelissen, Gunther; Kato, Fumiharu; Kool, Janne

doi:10.1007/s00208-014-1067-x

Cited by 39 publications

(56 citation statements)

References 57 publications

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“…We wish to emphasize that Theorem 1.2 does not follow from any classical simultaneous semistable reduction theorem, in that there exist finite morphisms ϕ : X → X of semistable models such that the inverse image of the corresponding skeleton of X is not a skeleton of X . (See, however, [24] where a skeletal version of Liu's theorem is derived from Liu's method of proof in the discretely valued case.) Our proof of Theorem 1.2 is entirely analytic, resting on an analysis of morphisms between open annuli and open balls; in particular, it makes almost no reference to semistable reduction theory and is therefore quite different from Liu and Liu-Lorenzini's approach.…”

Section: Skeletal Simultaneous Semistable Reductionmentioning

confidence: 97%

Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta

Amini

Baker

Brugallé

et al. 2015

Mathematical Sciences

186

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Let K be an algebraically closed, complete non-Archimedean field. The purpose of this paper is to carefully study the extent to which finite morphisms of algebraic K-curves are controlled by certain combinatorial objects, called skeleta. A skeleton is a metric graph embedded in the Berkovich analytification of X. A skeleton has the natural structure of a metrized complex of curves. We prove that a finite morphism of K-curves gives rise to a finite harmonic morphism of a suitable choice of skeleta. We use this to give analytic proofs of stronger 'skeletonized' versions of some foundational results of Liu-Lorenzini, Coleman, and Liu on simultaneous semistable reduction of curves. We then consider the inverse problem of lifting finite harmonic morphisms of metrized complexes to morphisms of curves over K. We prove that every tamely ramified finite harmonic morphism of -metrized complexes of k-curves lifts to a finite morphism of K-curves. If in addition the ramification points are marked, we obtain a complete classification of all such lifts along with their automorphisms. This generalizes and provides new analytic proofs of earlier results of Saïdi and Wewers. As an application, we discuss the relationship between harmonic morphisms of metric graphs and induced maps between component groups of Néron models, providing a negative answer to a question of Ribet motivated by number theory. This article is the first in a series of two. The second article contains several applications of our lifting results to questions about lifting morphisms of tropical curves.

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Section: Skeletal Simultaneous Semistable Reductionmentioning

confidence: 97%

Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta

Amini

Baker

Brugallé

et al. 2015

Mathematical Sciences

186

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“…This definition comes from [3]. Morphisms defined on graphs are sometimes indexed, as in [12]. In this paper, we will only consider non-indexed morphisms.…”

Section: Harmonic Morphisms Of Graphsmentioning

confidence: 99%

Graphs of gonality three

Aidun¹,

Dean²,

Morrison³

et al. 2019

Algebraic Combinatorics

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In 2013, Chan classified all metric hyperelliptic graphs, proving that divisorial gonality and geometric gonality are equivalent in the hyperelliptic case. We show that such a classification extends to combinatorial graphs of divisorial gonality three, under certain edge-and vertex-connectivity assumptions. We also give a construction for graphs of divisorial gonality three, and provide conditions for determining when a graph is not of divisorial gonality three.

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“…They give applications of their tropical Li-Yau inequality to uniform boundedness of torsion points on rank two Drinfeld modules, as well as to lower bounds on the modular degree of elliptic curves over function fields. The spectral bound from [CFK15] was subsequently refined by Amini and Kool in [AK14] to a spectral lower bound for the divisorial gonality (i.e., the minimal degree of a rank 1 divisor) of a metric graph Γ. In [AK14], as well as in the related paper [DJKM14], this circle of ideas is applied to show that the expected gonality of a random graph is asymptotic to the number of vertices.…”

Section: Further Readingmentioning

confidence: 99%