A logarithmic interpretation of Edixhoven's jumps for Jacobians

Eriksson, Dennis; Halle, Lars Halvard; Nicaise, Johannes

doi:10.1016/j.aim.2015.04.007

Cited by 8 publications

(3 citation statements)

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“…In the case where S is Dedekind this answers positively a question of Eriksson, Halle, and Nicaise in [EHN15].…”

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confidence: 66%

Models of Jacobians of curves

Holmes¹,

Molcho²,

Orecchia³

et al. 2020

Preprint

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We show that the Jacobians of prestable curves over toroidal varieties always admit Néron models. These models are rarely quasi-compact or separated, but we also give a complete classification of quasi-compact separated group-models of such Jacobians. In particular we show the existence of a maximal quasicompact separated group model, which we call the saturated model, which has the extension property for all torsion sections. The Néron model and the saturated model coincide over a Dedekind base, so the saturated model gives an alternative generalisation of the classical notion of Néron models to higher-dimensional bases; in the general case we give necessary and sufficient conditions for the Néron model and saturated model to coincide. The key result, from which most others descend, is that the logarithmic Jacobian of [MW18] is a log Neron model of the Jacobian.

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“…In the case where S is Dedekind this answers positively a question of Eriksson, Halle, and Nicaise in [EHN15].…”

mentioning

confidence: 66%

Models of Jacobians of curves

Holmes¹,

Molcho²,

Orecchia³

et al. 2020

Preprint

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“…It is difficult to construct examples beyond the case of elliptic curves because it is not well understood which weighted graphs may occur as skeleta of wildly ramified curves of genus ≥ 2. Nevertheless, basic calculations on abstract graphs indicate that one can get different limit measures as e(K /K) ranges over different residue classes modulo a suitable power of p. We believe that this phenomenon is related to fundamental ramification invariants introduced by Edixhoven [Ed92] and Chai and Yu [Ch00, CY01] and further studied in [EHN15].…”

Section: Further Questionsmentioning

confidence: 89%

Convergence of p-adic pluricanonical measures to Lebesgue measures on skeleta in Berkovich spaces

Jönsson¹,

Nicaise²

2020

Journal De L’École Polytechnique — Mathématiques

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Let K be a non-archimedean local field, X a smooth and proper K-scheme, and fix a pluricanonical form on X. For every finite extension K of K, the pluricanonical form induces a measure on the K -analytic manifold X(K ). We prove that, when K runs through all finite tame extensions of K, suitable normalizations of the pushforwards of these measures to the Berkovich analytification of X converge to a Lebesgue-type measure on the temperate part of the Kontsevich-Soibelman skeleton, assuming the existence of a strict normal crossings model for X. We also prove a similar result for all finite extensions K under the assumption that X has a log smooth model. This is a non-archimedean counterpart of analogous results for volume forms on degenerating complex Calabi-Yau manifolds by Boucksom and the first-named author. Along the way, we develop a general theory of Lebesgue measures on Berkovich skeleta over discretely valued fields. m K := (π K ) * |θ ⊗ K K | of the measure |θ ⊗ K K | to X an as K runs through finite extensions of K.Fix an algebraic closure K a of K, and let E a K be the set of finite extensions K of K in K a , ordered by inclusion. Let E ur K be the subset of E a K consisting of unramified extensions, and let K ur be the union of all extensions in E ur K (that is, the maximal unramified extension of K in K a ).Main Theorem. Assume X admits a log smooth model over the valuation ring R in K. Then there exist Lebesgue-type measures λ a and λ ur on X an , and positive constants c a K and c ur K for K in E a K and E ur K , respectively, such that lim K ∈E a K c a K m K = λ a , and lim K ∈E ur K c ur K m K = λ ur in the weak sense of positive Radon measures on X an .

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“…It follows easily from [Eriksson et al 2015, Proposition 3.3.4] that ω Ꮿ + /S + is isomorphic to ω Ꮿ/R (Ꮿ k,red + H ). We can copy the proofs of [Eriksson et al 2015, 3.3.2 and Proposition 3.3.6] to show that the coherent sheaf 1 Ᏸ + /S + on Ᏸ is perfect, so that we can define the canonical line bundle ω Ᏸ + /S + = det 1 Ᏸ + /S + on Ᏸ (the results in [Eriksson et al 2015] were formulated for H = 0 but the arguments carry over immediately). Since h is log étale, [Eriksson et al 2015, Proposition 3.3.6] also implies that we have a canonical isomorphism ω Ꮿ + /S + ∼ = h * ω Ᏸ + /S + .…”

Section: 33mentioning

confidence: 99%

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Algebra Number Theory

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A logarithmic interpretation of Edixhoven's jumps for Jacobians

Cited by 8 publications

References 20 publications

Models of Jacobians of curves

Models of Jacobians of curves

Convergence of p-adic pluricanonical measures to Lebesgue measures on skeleta in Berkovich spaces

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