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Arithmetic L-Functions and Differential Geometric Methods

Regulators IV, May 2016, Paris

herausgegeben von: Pierre Charollois, Gerard Freixas i Montplet, Vincent Maillot

Verlag: Springer International Publishing

Buchreihe : Progress in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book is an outgrowth of the conference “Regulators IV: An International Conference on Arithmetic L-functions and Differential Geometric Methods” that was held in Paris in May 2016. Gathering contributions by leading experts in the field ranging from original surveys to pure research articles, this volume provides comprehensive coverage of the front most developments in the field of regulator maps. Key topics covered are:

• Additive polylogarithms

• Analytic torsions

• Chabauty-Kim theory

• Local Grothendieck-Riemann-Roch theorems

• Periods

• Syntomic regulator

The book contains contributions by M. Asakura, J. Balakrishnan, A. Besser, A. Best, F. Bianchi, O. Gregory, A. Langer, B. Lawrence, X. Ma, S. Müller, N. Otsubo, J. Raimbault, W. Raskin, D. Rössler, S. Shen, N. Triantafi llou, S. Ünver and J. Vonk.

Inhaltsverzeichnis

Frontmatter

Regulators of K1 of Hypergeometric Fibrations

Abstract

We study a deformation of what we call hypergeometric fibrations. Its periods and K₁-regulators are described in terms of hypergeometric functions ₃F₂ in a variable given by the deformation parameter.

Masanori Asakura, Noriyuki Otsubo

Two Recent p-adic Approaches Towards the (Effective) Mordell Conjecture

Abstract

We give an introductory account of two recent approaches towards an effective proof of the Mordell conjecture, due to Lawrence–Venkatesh and Kim. The latter method, which is usually called the method of Chabauty– Kim or non-Abelian Chabauty in the literature, has the advantage that in some cases it has been turned into an effective method to determine the set of rational points on a curve, and we illustrate this by presenting three new examples of modular curves where this set can be determined.

J. S. Balakrishnan, A. J. Best, F. Bianchi, B. Lawrence, J. S. Müller, N. Triantafillou, J. Vonk

The Syntomic Regulator for K2 of Curves with Arbitrary Reduction

Abstract

We give a formula for the syntomic regulator on K₂ of a proper curve X over a p-adic field K. This generalizes the results of [Bes00c] where the curve was assumed to have good reduction. The formula is essentially the same with Coleman integration replaced by by Vologodsky integration.

Amnon Besser

Toric Regulators

Abstract

In the mid-19th century, Dirichlet (for quadratic fields) and then Dedekind defined a regulator map relating the units in the ring of integers of an algebraic number field of finite degree over Q with r₁ real embeddings and 2r₂complex embeddings to a lattice of codimension one in a Euclidean space of dimension r₁ + r₂.

Amnon Besser, Wayne Raskind

Higher Displays Arising from Filtered de Rham–Witt Complexes

Abstract

For a smooth projective scheme X over a ring R on which p is nilpotent that meets some general assumptions we prove that the crystalline cohomology is equipped with the structure of a higher display which is a relative version of Fontaine’s strongly divisible lattices. Frobenius-divisibility is induced by the Nygaard filtration on the relative de Rham–Witt complex. For a nilpotent PD-thickening S/R we also consider the associated relative display and can describe it explicitly by a relative version of the Nygaard filtration on the de Rham–Witt complex associated to a lifting of X over S. We prove that there is a crystal of relative displays if moreover the mod p reduction of X has a smooth and versal deformation space.

Oli Gregory, Andreas Langer

Orbifold Submersion and Analytic Torsions

Abstract

In this paper, we establish the curvature theorem of determinant line bundles for an orbifold Kähler fibration as an extension of Bismut–Gillet– Soulé’s curvature theorem. Then we introduce Bismut–Köhler analytic torsion form for an orbifold Kähler fibration. Finally we calculate the behaviour of the Quillen metric by orbifold submersions as an extension of Berthomieu– Bismut’s result.

Xiaonan Ma

Analytic Torsion, Regulators and Arithmetic Hyperbolic Manifolds

Abstract

This paper is a survey of some topics pertaining to the regulators associated to the homology of Riemannian manifolds. These were originally introduced by Ray–Singer in order to give a definition of Reidemeister torsion for a closed Riemannian manifold. This torsion was conjectured to be equal to the analytic torsion defined by purely analytic means, which was proven shortly afterwards by Cheeger and Müller. We will give a short account of this theorem before turning to our main theme of interest which concerns arithmetic groups and locally symmetric spaces.

In recent work of Bergeron–Venkatesh and others regulators have appeared as an obstruction in the study of torsion in the cohomology of arithmetic groups. In this context regulators also have an arithmetic significance, and this was explored further by Calegari–Venkatesh and Bergeron–şengün– Venkatesh. Finally, since many interesting arithmetic locally symmetric spaces are not compact it is natural to attempt to extend the definition of regulators, and the Cheeger–Müller theorem, to them. We will survey the work of Calegari–Venkatesh, Pfaff and Pfaff together with the author on this topic.

Jean Raimbault

A Local Refinement of the Adams–Riemann–Roch Theorem in Degree One

Abstract

We prove that the Adams–Riemann–Roch theorem in degree one (i.e., at the level of the Picard group) can be lifted to an isomorphism of line bundles, compatibly with base change.

Damian Rössler

Analytic Torsion and Dynamical Flow:A Survey on the Fried Conjecture

Abstract

Given an acyclic and unitarily flat vector bundle on a closed manifold, Fried conjectured an equality between the analytic torsion and the value at zero of the Ruelle zeta function associated to a dynamical flow. In this survey, we review the Fried conjecture for different flows, including the suspension flow, the Morse–Smale flow, the geodesic flow, and the Anosov flow.

Shu Shen

A Survey of the Additive Dilogarithm

Abstract

Borel’s construction of the regulator gives an injective map from the algebraic K–groups of a number field to its Deligne–Beilinson cohomology groups. This has many interesting arithmetic and geometric consequences. The formula for the regulator is expressed in terms of the classical polyogarithm functions. In this paper, we give a survey of the additive dilogarithm and the several different versions of the weight two regulator in the infinitesimal setting. We follow a historical approach which we hope will provide motivation for the definitions and the constructions.

Sinan Ünver

Titel: Arithmetic L-Functions and Differential Geometric Methods
herausgegeben von: Pierre Charollois
Gerard Freixas i Montplet
Vincent Maillot
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-65203-6
Print ISBN: 978-3-030-65202-9
DOI: https://doi.org/10.1007/978-3-030-65203-6

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Inhaltsverzeichnis

Frontmatter

Regulators of K1 of Hypergeometric Fibrations

Two Recent p-adic Approaches Towards the (Effective) Mordell Conjecture

The Syntomic Regulator for K2 of Curves with Arbitrary Reduction

Toric Regulators

Higher Displays Arising from Filtered de Rham–Witt Complexes

Orbifold Submersion and Analytic Torsions

Analytic Torsion, Regulators and Arithmetic Hyperbolic Manifolds

A Local Refinement of the Adams–Riemann–Roch Theorem in Degree One

Analytic Torsion and Dynamical Flow:A Survey on the Fried Conjecture

A Survey of the Additive Dilogarithm

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