Lecture Notes in Mathematics Lars Halvard Halle Johannes Nicaise. Néron Models and Base Change

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1 Lecture Notes in Mathematics 2156 Lars Halvard Halle Johannes Nicaise Néron Models and Base Change

2 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

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

5 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.

6 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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8 Contents Part I About This Book 1 Content of This Book Introduction Motivation and Background Abelian Varieties in Number Theory and Geometry Degenerating Families of Curves Néron Models Semi-Abelian Reduction Behaviour Under Base Change Jacobians, Stable Curves and Semi-Abelian Reduction Example: Elliptic Curves Motivic Zeta Functions Aim of This Book Semi-Abelian Varieties and Wildly Ramified Jacobians A Guiding Principle 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 Reductive Rank Néron Models The Néron Model and the Component Group The Toric Rank Néron Models and Base Change Example: The Néron lft-model of a Split Algebraic Torus vii

9 viii Contents The Néron Component Series Semi-Abelian Reduction Non-Archimedean Uniformization Models of Curves Sncd-Models and Combinatorial Data A Theorem of Winters Néron Models 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-Models and Tame Base Change Base Change and Normalization Local Computations Minimal Desingularization The Characteristic Polynomial and the Stabilization Index The Characteristic Polynomial The Stabilization Index Applications to sncd-models and Base 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 Smooth Sheaf 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 Uniformization of Semi-Abelian Varieties 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

10 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 Invariant Differential Forms Elliptic Curves Behaviour Under Non-Archimedean Uniformization Jumps of Jacobians Dependence on Reduction Data The Base Change Conductor and the Artin Conductor Some Comparison Results Algebraic Tori Saito s Discriminant-Conductor Formula Elliptic Curves The Potential Degree of 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 Functions of Jacobians Behaviour of the Identity Component Behaviour of the Order Function Rationality and Poles Rationality of the Zeta Function Poles and Monodromy Prym Varieties

11 x Contents 9 Cohomological Interpretation of the Motivic Zeta Function The Trace Formula for Semi-Abelian Varieties The Rational Volume The Trace Formula and the Number of Néron Components Cohomological Interpretation of the Motivic Zeta Function The Trace Formula for Jacobians The Monodromy Zeta Function The Trace Formula for Jacobians 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

12 Part I About This Book

13 Chapter 1 Content of This Book The aim of this book is to make a detailed analysis of the behaviour of Néron models of semi-abelian varieties under ramified base change, and apply the results to the study of motivic zeta functions. The main invariants we are interested in are the component group of the Néron model, Chai and Yu s base change conductor, and Edixhoven s filtration. In previous work, most notably [HN10, HN11a], we have considered the case of a tamely ramified abelian variety. Here we will extend these results to semi-abelian varieties and wildly ramified Jacobians. We will start with an overview of the chapters of this text. A brief summary can also be found at the beginning of each chapter. We denote by R a complete discrete valuation ring with quotient field K and separably closed residue field k. The characteristic exponent of k will be denoted by p. We fix a separable closure K s of K. In Chap. 2 we provide a detailed introduction to the book, with motivation and background on semi-abelian varieties, Néron models and motivic zeta functions. At the end of the chapter, the reader can find a list of notations that will be used throughout the book. Chapter 3 contains preliminary material on group schemes, models of curves and related topics. We recall key results from the literature and prove some basic new properties that will be needed in the remainder of the text. Part II is the longest part of this monograph; it is devoted to the study of Néron component groups of semi-abelian K-varieties. The component series of a semiabelian K-variety G is a generating series that encodes the orders of the component groups of G after base change to all tame finite extensions of K. One of our main objectives is to prove the rationality of the Néron component series and to determine the order of its pole at T D 1; this is the pole that influences the behaviour of the motivic zeta function. In Chap. 4, we investigate wildly ramified Jacobian varieties. Even in this situation, many of the methods we used for tamely ramified abelian varieties are no longer sufficient, or applicable. We will show that the behaviour of the Néron model of a Jacobian is controlled by a new invariant, which we call the Springer International Publishing Switzerland 2016 L.H. Halle, J. Nicaise, Néron Models and Base Change, Lecture Notes in Mathematics 2156, DOI / _1 3

14 4 1 Content of This Book stabilization index. The definition of the stabilization index of a K-curve and the study of its basic properties will occupy an important part of the chapter. The approach we take is to make use of the close relationship between Néron models of the Jacobian Jac.C/ of a K-curve C and regular models of C. More precisely, if we fix a regular, proper and flat R-model C of a smooth and proper K-curve C of index one, then the relative Picard scheme Pic 0 C =R is canonically isomorphic to the identity component A o of the Néron model A of the Jacobian A D Jac.C/; this is a fundamental theorem of Raynaud [Ra70]. Because of this link, many invariants associated to A can also be computed on C. In particular, this is true for the component group. A key technical step in Chap. 4 is therefore to provide a detailed description of the behaviour of regular models of K-curves under tame extensions of K. Assume that the special fiber C k is a strict normal crossings divisor. Then we will call C an sncd-model of C. For every positive integer d prime to p, we denote by K.d/ the unique degree d extension of K in K s, and by R.d/ its valuation ring. We write C d for the normalization of C R R.d/, andc.d/ for the minimal desingularization of C d. We will show that C.d/ is an sncd-model of C K K.d/ and explain how its structure can be determined from the structure of C.Inorderto obtain these results, we show that C d has at most locally toric singularities, which can be explicitly resolved using the results in Kiraly s PhD thesis [Ki10]. We provide an appendix with some basic results on the resolution of locally toric singularities, because this part of [Ki10] has not been published. Alternatively, one could use the language of logarithmic geometry [Ka94]. Next, we define the stabilization index e.c/.ifc is tamely ramified, it coincides with the degree of the minimal extension over which C acquires semi-stable reduction, but this is not true in general. The main property of e.c/ is that one can make a very precise comparison of the special fibers of C and C.d/ for any d prime to e.c/; see Proposition The key point is that the combinatorial structure of C.d/ k only depends on the combinatorial structure of C k, and not on the characteristic of k. This results allows us to extend certain facts from [HN10] from tamely ramified abelian varieties to wildly ramified Jacobians, which is crucial for proving the rationality of the component series. Our strategy is to use Winters theorem on the existence of curves with fixed reduction type to reduce to the case where K has equal characteristic zero; then every abelian K-variety is tamely ramified. It is surprisingly hard to give a direct combinatorial proof of our results on the component series, even for tamely ramified Jacobians. In Chap. 5, we switch our attention to Néron component groups of semi-abelian varieties. The main problem here is to understand the behaviour of the torsion part of the component group under finite extensions of the base field K. Our approach is based on two main tools. The first one is non-archimedean uniformization in the sense of [BX96]. This theory shows that, in the rigid analytic category, one can write every abelian K-variety as a quotient of a semi-abelian K-variety E by an étale lattice such that the abelian part of E has potential good reduction; in this way, one can try to reduce the study of Néron component groups to the case of tori and abelian varieties with potential good reduction. We extend

15 1 Content of This Book 5 this construction to semi-abelian varieties. The main complication is that, in this case, one needs to replace E by an analytic extension of a semi-abelian K-variety by a K-torus. This extension will usually not be algebraic, so that one cannot use the theory of Néron models for algebraic groups. Neither can one apply the theory of formal Néron models in [BS95], because the set of K-points on the extension can be unbounded. This brings us to the second tool in our approach: the sheaf-theoretic Néron model of Bosch and Xarles [BX96]. They interpret the formation of the Néron model as a pushforward operation on abelian sheaves from the smooth rigid site of SpK to the smooth formal site on Spf R, and show how one can recover the component group from this sheaf-theoretic interpretation. An advantage of this approach, which we already exploited in [HN10], is that one can quantify the lack of exactness of the Néron functor by means of the derived functors of the pushforward. A second advantage is that one can associate a sheaf-theoretic Néron model and a component group to any smooth abelian sheaf on SpK, and in particular to any commutative rigid K-group, even when a geometric (formal) Néron model does not exist. This is particularly useful when considering non-archimedean uniformizations of semi-abelian varieties as above. With these tools at hand, we can control the behaviour of the torsion part of the component group of a semi-abelian variety under finite extensions of the base field K. Our main result is Theorem , which says, in particular, that the component series of a semi-abelian K-variety G is rational if the abelian part G ab of G is tamely ramified or a Jacobian, and also if G ab has potential multiplicative reduction. In Part III, we study Chai s base change conductor of a semi-abelian variety, and some related invariants that were introduced by Edixhoven. These invariants play a key role in the determination of the poles of motivic zeta functions of semi-abelian varieties. In Chap. 6 we define a new invariant for wildly ramified semi-abelian K- varieties G, thetame base change conductor c tame.g/. This value is defined as the sum of the jumps (counting multiplicities) in Edixhoven s filtration on the special fiber of the Néron model of G. Equivalently, one can also define c tame.g/ as a limit of base change conductors with respect to all finite tame extensions of K in K s. If G is the Jacobian of a curve C, then we show that these jumps, and hence also c tame.g/, only depend on the combinatorial data of the special fiber of the minimal sncd-model of C. By construction, it is clear that c tame.g/ D c.g/ if G is tamely ramified. In the wild case, this equality may no longer hold. For instance, we will show that for an elliptic K-curve E, the equality c.e/ D c tame.e/ holds if and only if E is tamely ramified, and we interpret the error term in the wildly ramified case. It is natural to ask how c.g/ is related to other arithmetic invariants that one can associate to a semi-abelian K-variety G. Recall that, for algebraic tori, the main result of [CY01] states that the base change conductor of the torus equals one half of the Artin conductor of its cocharacter module. The situation is more delicate for abelian varieties: the base change conductor is not invariant under isogeny, so that it certainly cannot be computed from the Tate module of A with Q`-coefficients. In Chap. 7, westudythecasewhereg is the Jacobian of a curve C. A promising

16 6 1 Content of This Book candidate to consider is the so called Artin conductor Art.C/ of the curve C. This important invariant was introduced by S. Bloch and further studied by Saito in [Sa88]. It measures the difference between the Euler characteristics of the geometric generic and special fibers of the minimal regular model of C, with a correction term coming from the Swan conductor of the `-adic cohomology of C. It is not reasonable to expect that Art.C/ and c.jac.c// contain equivalent information, since Art.C/ vanishes if and only if C has good reduction while c.jac.c// vanishes if and only if C has semistable reduction. However, one can hope to express c.jac.c// in terms of Art.C/ and a suitable measure for the defect of potential good reduction of C. We will establish such an expression for curves of genus 1 or 2. For elliptic curves E, we get a clear picture: we will show that 12 c.e/ is equal to Art.E/ if E has potential good reduction, and to Art.E/ v K.j.E// else. In the latter case, one can view v K.j.E// as a measure for the defect of potential good reduction by noticing that it is equal to jˆ.e K L/j=ŒL W K, for any finite extension L of K such that E K L has semi-abelian reduction. Our method for genus 2 curves is somewhat indirect; in this case, the curve is hyperelliptic, which allows one to define a minimal discriminant min 2 K and its valuation v K. min /. We will compare v K. min / with the base change conductor c.jac.c//. The relationship to Art.C/ then becomes clear due to work of Liu [Li94] and Saito [Sa88], where Art.C/ and v. min / are compared. One should point out that already for genus 2, the invariants c.jac.c// and Art.C/ seem to diverge ; the base change conductor c.jac.c// rather seems to behave like v. min /.However,it is not clear how to generalize the definition of v. min / to curves of higher genus. We finally turn our focus to motivic zeta functions in Part IV. We recall the definition of the motivic zeta function Z G.T/ of a semi-abelian K-variety G in Chap. 8. It measures in a very fine way how the Néron model of G changes under tamely ramified base change. Our principal result, Theorem , extends the main theorem in [HN11a] to Jacobians and to tamely ramified semi-abelian varieties. More precisely, let G be a semi-abelian K-variety with abelian part G ab, and assume either that G is tamely ramified or that G is a Jacobian. Then we prove that Z G.T/ is a rational function and that Z G.L s / has a unique pole at s D c tame.g/ of order t tame.g ab / C 1.Heret tame.g ab / denotes the tame potential toric rank of G ab, that is, the maximum of the toric ranks of G ab K K 0 as K 0 runs over the finite tame extensions of K. We will also discuss similar results for Prym varieties. In Chap. 9, we establish a cohomological interpretation of the motivic zeta function Z G.T/ by means of a trace formula; this is similar to the Grothendieck-Lefschetz trace formula for varieties over finite fields and the resulting cohomological interpretation of the Hasse-Weil zeta function. Let ` be a prime different from p. Denote by K t thetameclosureofk in K s and choose a topological generator of the tame inertia group Gal.K t =K/. We denote by P G.T/ the characteristic polynomial of on the tame `-adic cohomology of G. One of our main results says that, if G has maximal unipotent rank, the prime-to-p part of the order of the Néron component group of G is equal to P G.1/. We deduce from this result that, for every tamely ramified semi-abelian K-variety G,the`-adic Euler characteristic of the special fiber of the Néron model of G is equal to the trace of on the `-adic cohomology of G.

17 1 Content of This Book 7 This yields a trace formula for the specialization of the motivic zeta function Z G.T/ with respect to the `-adic Euler characteristic. We also give an alternative proof of this result for a Jacobian Jac.C/ using a computation on the tame nearby cycles on an sncd-model C for the curve C. On the way, we recover a formula of Lorenzini for the characteristic polynomial P Jac.C/.T/ in terms of the geometry of C k. To conclude, we formulate some interesting open problems and directions for further research in Part V.

18 Chapter 2 Introduction 2.1 Motivation and Background Abelian Varieties in Number Theory and Geometry Abelian varieties play a central role in algebraic geometry and number theory. Historically, they are rooted in complex analysis, more precisely in the theory of elliptic integrals and abelian functions. In the first half of the nineteenth century, they appeared implicitly in the work of Abel and Jacobi as Jacobians of complex algebraic curves. The theory of abelian varieties over arbitrary fields was initiated by Weil in the 1940s as a part of his program to prove the Riemann Hypothesis for curves over finite fields. Abelian varieties also play a key role in class field theory, Faltings proof of the Mordell Conjecture and Wiles proof of Fermat s Last Theorem, to cite only a few of the manifold fundamental applications. Today, the study of the geometric and arithmetic properties of abelian varieties is an active and well-established field of research. We refer to [Mu08] for the basic theory, and to [Mi86] for a brief introduction. A fundamental question about geometric objects is their behaviour in families. In this book, we will mainly be concerned with families of abelian varieties over a regular base of dimension one, such as a smooth algebraic curve over a field (geometric setting) or the spectrum of the ring of integers in a number field (arithmetic setting). We will focus on the local behaviour of the family around a closed point of the base; if we pass to the localisation at this point, the base becomes the spectrum of a discrete valuation ring. The principal questions we will study are the following: given an abelian (or semi-abelian) variety over the quotient field of a discrete valuation ring R, what is the most natural way to extend it to a scheme over R? What can we say about the arithmetic and geometric properties of this extension? These questions lead to the theory of Néron models, which we will introduce below. Springer International Publishing Switzerland 2016 L.H. Halle, J. Nicaise, Néron Models and Base Change, Lecture Notes in Mathematics 2156, DOI / _2 9

19 10 2 Introduction Degenerating Families of Curves It is instructive to first recall a few classical results from the theory of curves. We fix an integer g 2. A key step in the study of algebraic curves is the construction of the moduli space M g of smooth, projective, geometrically connected curves of genus g. This moduli space is a connected smooth Deligne-Mumford stack over Z, but it is not proper: the problem is precisely that, in general, a smooth, projective, geometrically connected curve over a discretely valued field K cannot be extended to a smooth and proper scheme over the valuation ring (even after passing to a finite extension of K). For geometric applications (for instance, to do intersection theory) it is desirable to construct a compactification M g of M g such that the points on the boundary still have a geometric interpretation. More precisely, one would like the stack M g to represent families of (possibly singular) curves of a certain type.t/. These families should satisfy the following properties: (1) In order to get a representable functor, the class.t/ should be stable under base change. (2) In order to get a proper Deligne-Mumford stack, every smooth, projective, geometrically connected curve over a henselian discretely valued field K should extend uniquely to a family of type.t/ over the valuation ring of K, after passing to a finite extension of K (the finite extension is due to the fact that we work on a Deligne-Mumford stack, not a scheme; the henselian hypothesis guarantees that finite extensions of K are again discrete valuation rings). Such a class.t/ has been defined by Deligne and Mumford in their seminal paper [DeMu69]: we can take families of so-called stable curves. A stable curve over a field is a geometrically connected proper curve with at most nodal singularities and finite automorphism group. The functor of stable curves of arithmetic genus g is representable by a smooth and proper Deligne-Mumford stack M g that contains M g as a dense open substack, and the boundary is a strict normal crossings divisor whose strata represent the different combinatorial types of singular stable curves of arithmetic genus g. A key step towards this result is the Stable Reduction Theorem for curves, which states that the above property (2) holds for the class of stable curves Néron Models Now, we return to the case of abelian varieties. Let R be a discrete valuation ring with quotient field K and residue field k, andleta be an abelian variety over K. Again, it is not always possible to extend A to an abelian scheme over R, or even a smooth and proper R-scheme. One could ask for a canonical extension to a proper R-scheme, but in this case it is more natural to preserve the group structure on A. Néron has proved in [Né64] that there exists a canonical extension of A to a smooth separated

20 2.1 Motivation and Background 11 commutative group scheme A of finite type over R, known as the Néron model of A. Its distinguishing feature is the so-called Néron mapping property, which says that for every smooth R-scheme Z, every morphism of K-schemes Z R K! A extends uniquely to an R-morphism Z! A. Informally speaking, we can say that A is the minimal smooth R-model of A. For a detailed account of the construction and basic properties of Néron models, we refer to the excellent textbook [BLR90]. Néron models have become an invaluable tool for studying geometric and arithmetic properties of abelian varieties. One of Néron s prime motivations to construct these models was the theory of heights in diophantine geometry, which also played an essential role in Faltings proof of the Mordell Conjecture (every curve of genus at least two over a number field has only finitely many rational points). We will not say anything about heights on abelian varieties in this book, although they are closely related to some of the invariants that we will study. The basic idea is that we get a distinguished set of volume forms on A by taking the ones that extend to relative volume forms on the Néron model A, and one can put a natural metric on the canonical line bundle! A by declaring that any distinguished volume form has norm one Semi-Abelian Reduction Taking the special fiber of the Néron model allows one to define a canonical reduction A k D A R k of A.This is a smooth commutativegroupscheme overk, but the geometric structure of A k can be substantially more complicated than that of A. In general one cannot expect that the reduction of A is again an abelian variety; it may be non-proper and even disconnected. The shape of A k reflects the arithmetic properties of A over the field K (or, in geometric terms, the way in which A degenerates at the closed point of Spec R). We will now describe the principal invariants that one can attach to A k. Let Ak o be the identity component of A k. The group scheme ˆ.A/ D A k =A o k of connected components of A k is a finite étale commutative group scheme over k, known as the Néron component group of A. Ifk is separably closed, we can simply identify it with the constant group consisting of its k-points. The order of the component group is a subtle invariant of the abelian variety A. IfA is obtained by localizing an abelian variety over a number field at a prime p, then this order is known as the Tamagawa number at p. It appears in the Birch and Swinnerton- Dyer Conjecture, in the predicted formula for the leading coefficient of the Taylor expansion of the L-function at s D 1.

21 12 2 Introduction The identity component Ak o is canonically an extension of an abelian k-variety B by a commutative linear algebraic k-group G; this is the so-called Chevalley decomposition of Ak o.ifk is perfect, one can show that G is the product of a torus T and a unipotent group U, whose dimensions are called the toric and unipotent rank of A, respectively. In the special case where U Df0g, one says that A has semiabelian reduction. In the literature, one often uses the term semistable reduction instead, but we prefer to avoid it because it might lead to confusion with semistable models of K-varieties. The most important structural result concerning Néron models is Grothendieck s Semistable Reduction Theorem [SGA7-I, IX.3.6]. Assuming that R is strictly henselian and fixing a separable closure K s of K, it asserts the existence of a unique minimal finite extension L of K in K s such that A K L has semi-abelian reduction. We say that A is tamely ramified if L is a tame extension of K (that is, of ramification index prime to the characteristic exponent of k), and wildly ramified else Behaviour Under Base Change The Néron model is functorial in A, but otherwise it does not have good functorial properties. A first problem is that it behaves poorly in exact sequences. Another important complication is that its formation does not commute with ramified base change. Assume that K is henselian. Let K 0 be a finite extension of K and denote by R 0 the integral closure of R in K 0. This is again a discrete valuation ring. We set A 0 D A K K 0, and we denote by A 0 the Néron model of A 0.SinceA R R 0 is smooth over R 0, the Néron mapping property of A 0 implies that there exists a unique morphism of R 0 -group schemes h W A R R 0! A 0 that extends the canonical isomorphism on generic fibers. If R 0 is unramified over R then A 0 is smooth over R so that we can use the Néron mapping property of A to produce an inverse for h; thus h is an isomorphism. This fails if R 0 is ramified over R. If A has semi-abelian reduction, then h is an open immersion; in particular, it is an isomorphism on identity components (but the number of components can grow). This property underlies the importance of Grothendieck s Semistable Reduction Theorem. In the general case, however, it is quite hard to describe the properties of the base change morphism h, especially when A is wildly ramified. This is the main objective of this book.

22 2.1 Motivation and Background Jacobians, Stable Curves and Semi-Abelian Reduction There is a close connection between semi-abelian reduction of abelian varieties and the Stable Reduction Theorem for curves that we recalled above. Let C be a smooth, projective, geometrically connected curve of genus 2 over K, andleta be its Jacobian variety. Then one can use models of C over the valuation ring R to describe the Néron model of A. More precisely, Raynaud proved in [Ra70] thatifc =R is a proper, flat and regular model of C, then, under certain mild assumptions, the relative Picard scheme Pic 0 C =R is isomorphic to the identity component A 0 of the Néron model A of A. From this, one can deduce that A has semi-abelian reduction if and only if C has a stable model over R (see [DeMu69, 2.4]). In fact, Deligne and Mumford used Raynaud s result to deduce the Stable Reduction Theorem for curves from Grothendieck s Semistable Reduction Theorem for abelian varieties Example: Elliptic Curves As an illustration, we will describe the case of elliptic curves. Detailed proofs can be foundin Chap. IV of [Si94]. For simplicity, we assume that R is a henselian discrete valuation ring with quotient field K and algebraically closed residue field k. Let E=K be an elliptic curve defined by the affine Weierstrass equation y 2 C.a 1 x C a 3 /y D x 3 C a 2 x 2 C a 4 x C a 6 ; (2.1.1) where a i 2 K for each i. The homogenization of this equation describes E as a smooth cubic in P 2 K, where the identity point O 2 E.K/ for the group law on E has homogeneous coordinates.0 W 1 W 0/. ToEq.(2.1.1) one can associate the discriminant 2 ZŒa 1 ;:::;a 6. Consequently, is an element of K, and it is a basic fact that smoothness of E is equivalent to the non-vanishing of. After a suitable change of coordinates, it is possible to arrange that all coefficients a i belong to R. Thus Eq. (2.1.1) defines a family W of cubics in P 2 R, with generic fiber W R K Š E, and the discriminant is now actually an element of R. By performing further coordinate transformations if necessary, one can assume that the valuation of is minimal. The scheme W is then said to be a minimal Weierstrass model of E over R. Moreover, the group structure on E extends uniquely to the R-smooth locus W sm of W. The behaviour of W under reduction falls into three separate cases. First, when is a unit in R, the reduction W k D W R k P 2 k

23 14 2 Introduction is again an elliptic curve and W forms an abelian scheme over R. One says that E has good reduction. In this case, it is elementary to see that W is the Néron model of E (assuming that we already know that the Néron model exists). If is not a unit, one can show that the plane cubic curve W k has a unique singular point q, and now precisely two situations can occur: (1) W k has a node at q,and W sm k D W k nfqg ŠG m : In this case, one says that E has multiplicative reduction. (2) W k has a cusp at q,and W sm k D W k nfqg ŠG a : Then one says that E has additive reduction. Even though W sm is a smooth group scheme extending E over R, it can fail to satisfy the Néron mapping property. In order to construct the Néron model of E, it may therefore be necessary to modify W somewhat. Let X! W be the minimal desingularization of W. It is obtained by successively blowing up closed points in the special fiber lying over q, in a finite number of steps. The surface X is in fact the minimal regular model of the curve E. By removing the singular locus of X k, one obtains a smooth R-scheme E WD X n Sing.X k /; which, in the end, turns out to be the Néron model of E. We remark that W sm can in a natural way be identified with a subgroup scheme of E ; it is precisely the identity component E ı of E. We conclude this discussion of elliptic curves with some remarks about the behaviour of the Néron model under ramified extension of K. LetK 0 =K be a finite separable extension of degree n >1and let R 0 be the valuation ring of K 0.Firstof all, if E has good reduction over R, and hence the Néron model E is proper over R, then E R R 0 is again the Néron model of E K K 0. However, also the property of having multiplicative reduction is stable under ramified extensions. Indeed, in this situation it can be seen that the special fiber X k forms a reduced cycle of smooth rational curves, unless X D W, in which case X k is a plane cubic with a unique node. In any case, the base change X R R 0 is normal, with A n -singularities located exactly at the nodes in the special fiber. Let X 0! X R R 0 be the minimal desingularization. The exceptional locus over each singular point forms a chain of n 1 smooth rational curves. Thus, the special fiber Xk 0 is again a reduced cycle of smooth rational curves. Moreover, the Néron model E 0 of E K K 0 coincides with the R 0 -smooth locus of X 0, and so the identity component of Ek 0 is again isomorphic to G m. Note however that the total number of

24 2.1 Motivation and Background 15 connected components of has increased by a factor n D ŒK 0 W K when passing from E k to Ek 0. When E has additive reduction over R, the special fiber X k is again a configuration of rational curves, but now some of the components might appear with multiplicity greater than one in the divisor X k on X.Néron[Né64] classified all combinatorial possibilities for such configurations, and obtained the same list as Kodaira, who had previously classified the possible degenerate fibers in a minimal elliptic fibration over a complex disk. The behaviour of the Néron model under ramified extension of K is less straightforward to control in the case of additive reduction. In fact, if E has additive reduction, and K 0 =K is a ramified extension, E K K 0 might continue to have additive reduction, or the reduction type could change. However, by the Semistable Reduction Theorem for abelian varieties, one can always find a sufficiently large extension of K over which E acquires either good or multiplicative reduction Motivic Zeta Functions The problem of describing how Néron models behave under base extensions lies at the heart of our work on motivic zeta functions of abelian varieties. In order to explain this notion, we first need to introduce some notation. Let N 0 be the set of positive integers not divisible by the characteristic of k. For each d 2 N 0 we denote by K.d/ the unique degree d extension of K in K s. We put A.d/ WD A K K.d/ and we denote by A.d/ the Néron model of A.d/. In[HN11a], we defined the motivic zeta function Z A.T/ as Z A.T/ D X d2n 0 ŒA.d/ k L ord A.d/ T d 2 K 0.Var k /ŒŒT : Here K 0.Var k / denotes the Grothendieck ring of k-varieties, L denotes the class ŒA 1 k of the affine line in K 0.Var k /, and ord A is a function from N 0 to N whose definition will be recalled in Chap. 8. One can roughly say that Z A.T/ is the generating series for the reductions A.d/ k (up to a certain scaling by L), and thus encodes in a very precise way how the Néron model of A changes under tamely ramified extensions of K. Moreover, one can view this object as an analog of Denef and Loeser s motivic zeta function for complex hypersurface singularities. This link will not be pursued further in this monograph; we refer to [HN12] for a detailed survey, and an overview of the literature. Since each of the connected components of A.d/ k is isomorphic to the identity component A.d/ o k, we have the relation ŒA.d/ k Djˆ.A.d//jŒA.d/ o k

25 16 2 Introduction in K 0.Var k / for every d 2 N 0. Because of this fact, many properties of Z A.T/, such as rationality and the nature of its poles, are closely linked to analogous properties of the Néron component series SÂ.T/ D X d2n 0 jˆ.a.d//jt d 2 ZŒŒT that we introduced in [HN10] (there it was denoted S.AI T/). This series measures how the number of Néron components varies under tame extensions of K. Wewere able to prove in [HN10, 6.5] that it is a rational function when A is tamely ramified or A has potential multiplicative reduction. This was a key ingredient of our proof that the motivic zeta function Z A.T/ is rational if A is tamely ramified [HN11a]. Moreover, setting T D L s and viewing s as a formal variable, we showed that Z A.L s / has a unique pole. Interestingly, this pole coincides with an important arithmetic invariant of the abelian variety A: the base change conductor c.a/,which was introduced for tori by Chai and Yu [CY01] and for semi-abelian varieties by Chai [Ch00]. It is a nonnegative rational number that measures the defect of semiabelian reduction of A. 2.2 Aim of This Book Semi-Abelian Varieties and Wildly Ramified Jacobians One of the main purposes of this monograph is to extend the above mentioned results beyond the case of tamely ramified abelian varieties. For one thing, it is natural to ask what can be said without the tameness assumption. In general, it is not even clear if Z A.T/ is rational if A is wildly ramified. We establish this fact for Jacobians in Chap. 8. It remains a considerable challenge to understand the motivic zeta function of a wildly ramified abelian variety that is not a Jacobian, and likewise, to understand the Néron component series of a wildly ramified abelian variety that is not a Jacobian and does not have potential multiplicative reduction. Going in a different direction, one can ask about the situation for more general group schemes than abelian varieties. In [HN11a] we developed the general theory of motivic zeta functions to also include, in particular, the class of semi-abelian varieties. The existence of Néron models of semi-abelian K-varieties was proven in [BLR90]. An important difference with the case of abelian varieties is that the Néron model of a semi-abelian variety will, in general, only be locally of finite type. In order to get a meaningful definition of the motivic zeta function, one has to consider the maximal quasi-compact open subgroup scheme of the Néron model. On the level of component groups, this means that we consider the torsion subgroup of the component group.

26 2.2 Aim of This Book 17 While the methods we developed to study the behaviour of component groups of abelian varieties under base change can easily be extended to semi-abelian varieties, the torsion subgroup is a much more subtle invariant. An important complication is the lack of a geometric characterization of this object. A natural candidate would be the following: the Néron component group of the maximal split subtorus of a semiabelian K-variety G is a lattice of maximal rank inside the component group ˆ.G/, and one might hope to capture the torsion part by showing that this injection is split. We will show that this is usually not the case. For algebraic tori one can encompass this problem by passing to the dual torus, but no similar technique seems to exist for general semi-abelian varieties. Our approach consists in defining a suitable notion of non-archimedean uniformization for semi-abelian varieties. However, the uniformization space will no longer be an algebraic object, so that the existence of a (formal) Néron model is no longer guaranteed; instead, we will make a careful study of the properties of the sheaf-theoretic Néron model defined by Bosch and Xarles [BX96] A Guiding Principle One of the basic ideas in our work on component series and motivic zeta functions of abelian varieties is the expectation that the Néron model of an abelian K-variety A changes as little as possible under a tame extension K 0 =K that is sufficiently orthogonal to the minimal extension L=K where A acquires semi-abelian reduction. This principle was a crucial ingredient in establishing rationality and determining the poles of the component series SÂ.T/ and the motivic zeta function Z A.T/ in [HN10] and[hn11a]. What do we mean by as little as possible? It is unreasonable to require that the base change morphism h W A R R 0! A 0 is an isomorphism. Even when A has semi-abelian reduction, the number of connected components of the Néron model might still change; the rate of growth is determined by the toric rank t.a/ of A. It can be shown by elementary examples (with elliptic curves for instance) that if A does not have semi-abelian reduction, we cannot even require h to be an open immersion. The best we can ask for is that the following two properties are satisfied. (1) The number of components grows as if A had semi-abelian reduction: if K 0 =K is a tame extension of degree d,then jˆ.a K K 0 /jdd t.a/ jˆ.a/j:

27 18 2 Introduction (2) The k-varieties Ak o and.a 0 / o k define the same class in K 0.Var k /.By[Ni11b, 3.1], this is equivalent to the property that Ak o and.a 0 / o k have the same unipotent and reductive ranks, and isomorphic abelian quotients in the Chevalley decomposition. The meaning of sufficiently orthogonal is less clear. The most natural guess is that the extensions K 0 and L should be linearly disjoint over K, which is equivalent to asking that ŒK 0 WK and ŒLWK are coprime because the extension K 0 =K is tame. We have shown in [HN10] and[hn11a] that this condition is indeed sufficient when A is tamely ramified or A has potential multiplicative reduction. However, we will see that, for the Jacobian Jac.C/ of a K-curve C, the condition has to be modified: one needs to replace the degree of L over K by another invariant e.c/ that we call the stabilization index of C. It is defined in terms of the geometry of the R-models of the curve C. For general wildly ramified abelian K-varieties, it is not even clear if a suitable notion of orthogonality exists; we will come back to this problem in Part V Notation ( ) We denote by K a complete discretely valued field with ring of integers R and residue field k. We denote by m the maximal ideal of R, and by v K W K! Z [f1g the normalized discrete valuation on K. We assume that k is separably closed and we denote by p the characteristic exponent of k. The letter ` will always denote a prime different from p. For most of the results in this book, the conditions that R is complete and k separably closed are not serious restrictions: since the formation of Néron models commutes with extensions of ramification index one [BLR90, ], one can simply pass to the completion of a strict henselization of R. For some of the results we present, we need to assume that k is perfect (and thus algebraically closed); this will be clearly indicated at the beginning of the chapter or section. ( ) We fix a uniformizing parameter in R and a separable closure K s of K. We denote by K t thetameclosureofk in K s, and by P D Gal.K s =K t / the wild inertia subgroup of the inertia group I D Gal.K s =K/ of K. ( ) We denote by N 0 the set of positive integers prime to p.foreveryd 2 N 0, there exists a unique degree d extension of K inside K t, which we denote by K.d/. It is obtained by joining a d-th root of to K. The integral closure R.d/ of R in K.d/ is again a complete discrete valuation ring. For every K-scheme Y, we put Y.d/ D Y K K.d/. ( ) We will consider the special fiber functor./ k W.Schemes=R/!.Schemes=k/ W X 7! X k D X R k

28 2.2 Aim of This Book 19 as well as the generic fiber functor./ K W.Schemes=R/!.Schemes=K/ W X 7! X K D X R K: We will use the same notations for the special fiber functor from the category of formal R-schemes locally topologically of finite type to the category of k-schemes locally of finite type, resp. Raynaud s generic fiber functor from the category of formal R-schemes locally topologically of finite type to the category of quasiseparated rigid K-varieties. ( ) We denote by./ an the rigid analytic GAGA functor from the category of K-schemes of finite type to the category of quasi-separated rigid K-varieties. From now on, all rigid K-varieties will tacitly be assumed to be quasi-separated. ( ) For every separated k-scheme of finite type X, we denote by.x/ its `- adic Euler characteristic.x/ D X. 1/ i dim Hc i.x; Q`/: i0 It is well known that the value.x/ does not depend on `: ifk is finite it is a consequence of the cohomological interpretation of the Hasse-Weil zeta function, and the general case follows from a spreading-out argument to reduce to the case of a finite base field. ( ) For every field F, an algebraic F-group is a group scheme of finite type over F. A semi-abelian F-variety is a smooth commutative algebraic F-group that is an extension of an abelian F-variety by an algebraic F-torus. ( ) For every finitely generated abelian group ˆ, we denote by ˆtors the subgroup of torsion elements and by ˆfree the free abelian group ˆ=ˆtors.These operations define functors from the category of finitely generated abelian groups to the category of finite abelian groups and the category of free abelian groups of finite rank, respectively. Acknowledgements The second author was supported by the Fund for Scientific Research Flanders (G ) and the ERC Starting Grant MOTZETA (project ) of the European Research Council.

29 Chapter 3 Preliminaries 3.1 Galois Theory of K In this section, we assume that k is algebraically closed The Artin Conductor ( ) Let K 0 be a tame finite extension of K of degree d. Denote by R 0 the integral closure of R in K 0, and by m 0 the maximal ideal of R 0. Then the quotient m 0 =.m 0 / 2 is a rank one vector space over k, and the left action of D Gal.K 0 =K/ on K 0 induces an injective group morphism! Aut k.m 0 =.m 0 / 2 / D k whose image is the group d.k/ of d-th roots of unity in k. Thus we can identify Gal.K 0 =K/ and d.k/ in a canonical way. ( ) If L is a finite Galois extension of K with Galois group D Gal.L=K/, then the lower numbering ramification filtration. i / i 1 on is defined in [Se68, IV 1]. We quickly recall its definition. Denote by R L the valuation ring of L, bym L its maximal ideal and by L a uniformizer of L. Foreveryi 1, the subgroup i consists of all elements in such that v L. L. L // i C 1: Springer International Publishing Switzerland 2016 L.H. Halle, J. Nicaise, Néron Models and Base Change, Lecture Notes in Mathematics 2156, DOI / _3 21