Igusa Integrals and Volume Asymptotics in Analytic and Adelic Geometry

Chambert-Loir, Antoine; Tschinkel, Yuri

doi:10.1142/s1793744210000223

Cited by 69 publications

(101 citation statements)

References 39 publications

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“…In step (II), we establish an asymptotic formula for integral [P94], we show that f (s) has a meromorphic continuation beyond the first pole. Then the asymptotic formula for v(t) follows via a Tauberian argument (see also [CT10] for a similar argument). The completion of the argument proving Theorem 9.3 is different and requires several arguments.…”

Section: Then the Normalized Sampling Operatorsmentioning

confidence: 95%

Quantitative ergodic theorems and their number-theoretic applications

Gorodnik

Nevo

2014

Bull. Amer. Math. Soc.

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Abstract. We present an account of some recent applications of ergodic theorems for actions of algebraic and arithmetic groups to the solution of natural problems in Diophantine approximation and number theory. Our approach is based on spectral methods utilizing the unitary representation theory of the groups involved. This allows the derivation of ergodic theorems with a rate of convergence, an important phenomenon which does not arise in classical ergodic theory. Combining spectral and dynamical methods, quantitative ergodic theorems give rise to new and previously inaccessible applications. We demonstrate the remarkable diversity of such applications by deriving general uniform error estimates in non-Euclidean lattice points counting problems, explicit estimates in the sifting problem for almost-prime points on symmetric varieties, best-possible bounds for exponents of intrinsic Diophantine approximation on homogeneous algebraic varieties, and quantitative results on fast distribution of dense orbits on compact and non-compact homogeneous spaces.

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Section: Then the Normalized Sampling Operatorsmentioning

confidence: 95%

Quantitative ergodic theorems and their number-theoretic applications

Gorodnik

Nevo

2014

Bull. Amer. Math. Soc.

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“…This has been done in [5] for equivariant compactifications of additive groups G n a ; the same approach works here as well. We regard the height integrals as geometric versions of Igusa's integrals (see [6]). …”

Section: Introductionmentioning

confidence: 99%

Height Zeta Functions of Equivariant Compactifications of Unipotent Groups

Shalika

Tschinkel

2015

Comm Pure Appl Math

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“…These results should be viewed as a geometric version of our convergence statements for non-archimedean local fields. Closely related results over p-adic fields had already appeared in the literature in a somewhat different setting, especially in the work of Chambert-Loir and Tschinkel [CLT10].…”

Section: Related Resultsmentioning

confidence: 81%

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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Igusa Integrals and Volume Asymptotics in Analytic and Adelic Geometry

Cited by 69 publications

References 39 publications

Quantitative ergodic theorems and their number-theoretic applications

Quantitative ergodic theorems and their number-theoretic applications

Height Zeta Functions of Equivariant Compactifications of Unipotent Groups

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

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