Lecture Notes in Mathematics 2156

Transcription

1 Lecture Notes in Mathematics 2156 Editors-in-Chief: J.-M. Morel, Cachan B. Teissier, Paris Advisory Board: Camillo De Lellis, Zurich Mario di Bernardo, Bristol Alessio Figalli, Austin Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gabor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, Paris and NY Catharina Stroppel, Bonn Anna Wienhard, Heidelberg

2 More information about this series at

3 Lars Halvard Halle Johannes Nicaise Néron Models and Base Change 123

4 Lars Halvard Halle Dept. of Math. Sciences University of Copenhagen Copenhagen, Denmark Johannes Nicaise Dept. of Mathematics Imperial College London London, United Kingdom ISSN ISSN (electronic) Lecture Notes in Mathematics ISBN ISBN (ebook) DOI / Library of Congress Control Number: Mathematics Subject Classification (2010): 14K15, 14H40, 14G22, 14E18 Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by SpringerNature The registered company is Springer International Publishing AG Switzerland.

5 Preface We study various aspects of the behaviour of Néron models of semi-abelian varieties under finite extensions of the base field, with a special emphasis on wildly ramified Jacobians. In Part I, we analyse the behaviour of the component groups of Néron models, and we prove rationality results for a certain generating series encoding their orders. In Part II, we discuss Chai s base change conductor and Edixhoven s filtration and their relation to the Artin conductor. All of these results are applied in Part III to the study of motivic zeta functions of semi-abelian varieties. Part IV contains some intriguing open problems and directions for further research. The main tools in this work are non-archimedean uniformization and a detailed analysis of the behaviour of regular models of curves under base change. Copenhagen, Denmark London, UK Lars Halvard Halle Johannes Nicaise v

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7 Contents Part I About This Book 1 Content of This Book Introduction Motivation and Background Abelian Varieties in NumberTheoryandGeometry Degenerating Families of Curves NéronModels Semi-AbelianReduction BehaviourUnderBase Change Jacobians, Stable Curves and Semi-Abelian Reduction Example: Elliptic Curves Motivic Zeta Functions Aim ofthis Book Semi-Abelian Varieties and Wildly Ramified Jacobians A GuidingPrinciple Notation Preliminaries Galois Theory of K The Artin Conductor Isolating the Wild Part of the Conductor Subtori of Algebraic Groups Maximal Subtori Basic Properties of the ReductiveRank NéronModels The Néron Model and the Component Group The Toric Rank NéronModels and Base Change Example: The Néron lft-model of a Split AlgebraicTorus vii

8 viii Contents The Néron Component Series Semi-AbelianReduction Non-ArchimedeanUniformization Models of Curves Sncd-ModelsandCombinatorialData A Theoremof Winters NéronModels of Jacobians Semi-Stable Reduction Part II Néron Component Groups of Semi-Abelian Varieties 4 Models of Curves and the Néron Component Series of a Jacobian Sncd-ModelsandTame Base Change Base Change andnormalization Local Computations Minimal Desingularization The Characteristic Polynomial and the Stabilization Index The Characteristic Polynomial The Stabilization Index Applications to sncd-modelsandbase Change The Néron Component Series of a Jacobian Rationality of the Component Series Appendix:Locally Toric Rings Resolution of Locally Toric Singularities Tame Cyclic Quotient Singularities Component Groups and Non-Archimedean Uniformization Component Groups of Smooth Sheaves The Work of Bosch and Xarles Identity Component and Component Group of a SmoothSheaf Some Basic Properties of the Component Group The Trace Map The Index of a Semi-Abelian K-Variety Definition of the Index Example: The Index of a K-Torus Component Groups and Base Change Uniformizationof Semi-AbelianVarieties Bounded Rigid Varieties and Torsors Under Analytic Tori Behaviour of the Component Group Under Base Change The Component Series of a Semi-Abelian Variety... 85

9 Contents ix Part III Chai and Yu s Base Change Conductor and Edixhoven s Filtration 6 The Base Change Conductor and Edixhoven s Filtration Basic Definitions The Conductor of a Morphism of Modules The Base Change Conductor of a Semi-Abelian Variety Jumps and Edixhoven s Filtration Computing the Base Change Conductor InvariantDifferentialForms Elliptic Curves BehaviourUnderNon-ArchimedeanUniformization Jumps of Jacobians Dependenceon ReductionData The Base Change Conductor and the Artin Conductor Some ComparisonResults AlgebraicTori Saito s Discriminant-Conductor Formula Elliptic Curves The Potential Degreeof Degeneration A Formula for the Base Change Conductor Genus Two Curves Hyperelliptic Equations Minimal Equations Comparison of the Base Change Conductor and the Minimal Discriminant Part IV Applications to Motivic Zeta Functions 8 Motivic Zeta Functions of Semi-Abelian Varieties The Motivic Zeta Function Definition Decomposing the Identity Component Motivic Zeta Functionsof Jacobians Behaviour of the Identity Component Behaviourof the Order Function Rationality and Poles Rationality of the Zeta Function Poles and Monodromy Prym Varieties

10 x Contents 9 Cohomological Interpretation of the Motivic Zeta Function The Trace FormulaforSemi-AbelianVarieties The Rational Volume The Trace Formula and the Number of Néron Components Cohomological Interpretation of the Motivic Zeta Function The Trace FormulaforJacobians The Monodromy Zeta Function The Trace FormulaforJacobians Part V Some Open Problems 10 Some Open Problems The Stabilization Index The Characteristic Polynomial The Motivic Zeta Function and the Monodromy Conjecture Base Change Conductor for Jacobians Component Groups of Jacobians References