Complex multiplication and lifting problems. Ching-Li Chai Brian Conrad Frans Oort

Transcription

1 Complex multiplication and lifting problems Ching-Li Chai Brian Conrad Frans Oort Department of Mathematics, University of Pennsylvania, Philadelphia, PA address: Institute of Mathematics, Academia Sinica, 6F Astronomy-Mathematics Buliding, No. 1 Sec. 4 Roosevelt Rd., Taipei 19617, Taiwan address: chai@math.sinica.edu.tw Department of Mathematics, Stanford University, Stanford, CA address: conrad@math.stanford.edu Mathematisch Instituut, Budapestlaan 6, Postbus , Utrecht 3508 TA, The Netherlands address: f.oort@uu.nl

2 1991 Mathematics Subject Classification. Primary 14K10, 14G35; Secondary 14L05, 14F30, 11F32 Key words and phrases. abelian variety, complex multiplication, CM type, isogeny, moduli, Newton polygon, p-divisible group, reflex field, slopes

3 This book is dedicated to John Tate for what he taught us, and for his inspiration

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5 Preface During the Workshop on Abelian Varieties in Amsterdam in May 2006, the three authors of this book formulated two refined versions of a problem concerning lifting into characteristic 0 for abelian varieties over a finite field. These problems address the phenomenon of CM lifting : the lift into characteristic 0 is required to be a CM abelian variety (in the sense defined in ). The precise formulations appear at the end of Chapter 1 (see 1.8.5), as problems (I) and (IN). Abelian surface counterexamples to (IN) were found at that time: see , and see for a more thorough analysis. To our surprise, the same counterexamples (typical among toy models as defined in 4.1.3) play a crucial role in the general solution to problems (I) and (IN). This book is the story of our adventure guided by CM lifting problems. Ching-Li Chai thanks Hsiao-Ling for her love and support during all these years. He also thanks Utrecht University for hospitality during many visits, including the May 2006 Spring School on Abelian Varieties which concluded with the workshop in Amsterdam. Support by NSF grants DMS , DMS , and DMS is gratefully acknowledged. Brian Conrad thanks the many participants in the CM seminar at the University of Michigan for their enthusiasm on the topic of complex multiplication, as well as Columbia University for its hospitality during a sabbatical visit, and gratefully acknowledges support by NSF grants DMS , DMS , and DMS Frans Oort thanks the University of Pennsylvania for hospitality and stimulating environment during several visits. We are also grateful to Burcu Baran, Bas Edixhoven, Johan de Jong, Ofer Gabber, Ben Moonen, James Parson, and René Schoof for insightful and memorable discussions. vii

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7 Contents Preface vii Introduction 1 References 7 Notation and terminology 8 Chapter 1. Algebraic theory of complex multiplication Introduction Simplicity, isotypicity, and endomorphism algebras Complex multiplication Dieudonné theory and p-divisible groups CM types Abelian varieties over finite fields A theorem of Grothendieck and a construction of Serre CM lifting questions 78 Chapter 2. CM lifting over a discrete valuation ring Introduction Existence of CM lifting up to isogeny CM lifting to a normal domain up to isogeny: counterexamples Algebraic Hecke characters Theory of complex multiplication Local methods 122 Chapter 3. CM lifting of p-divisible groups Motivation and background Properties of a-numbers Isogenies and duality Some p-divisible groups with small a-number Earlier non-liftability results and a new proof A lower bound on the field of definition Complex multiplication for p-divisible groups An upper bound for a field of definition Appendix: algebraic abelian p-adic representations of local fields Appendix: questions and examples on extending isogenies 180 Chapter 4. CM lifting of abelian varieties up to isogeny Introduction Classification and Galois descent by Lie types 201 ix

8 x CONTENTS 4.3. Tensor construction for p-divisible groups Self-duality and CM lifting Striped and supersingular Lie types Complex conjugation and CM lifting 231 Appendix A. Some arithmetic results for abelian varieties 241 A.1. The p-part of Tate s work 241 A.2. The Main Theorem of Complex Multiplication 250 A.3. A converse to the Main Theorem of Complex Multiplication 285 A.4. Existence of algebraic Hecke characters 289 Appendix B. CM lifting via p-adic Hodge theory 313 B.1. A generalization of the toy model 313 B.2. Construct CM lifting by p-adic Hodge theory 326 B.3. Dieudonné theories over a perfect field of characteristic p 337 B.4. p-adic Hodge theory and a formula for the closed fiber 354 Bibliography 365 Notes on Quotes 371 Glossary of Notations 373 Index 379

9 Introduction I restricted myself to characteristic zero: for a short time, the quantum jump to p 0 was beyond the range... but it did not take me too long to make this jump. Oscar Zariski The arithmetic of abelian varieties with complex multiplication over a number field is fascinating. However this will not be our focus. We study the theory of complex multiplication in mixed characteristic. Abelian varieties over finite fields. In 1940 Deuring showed that an elliptic curve over a finite field can have an endomorphism algebra of rank 4 [31, 2.10]. For an elliptic curve in characteristic zero with an endomorphism algebra of rank 2 (rather than rank 1, as in the generic case), the j-invariant is called a singular j- invariant. For this reason elliptic curves with even more endomorphisms, in positive characteristic, are called supersingular. 1 Mumford observed as a consequence of results of Deuring that for any elliptic curves E 1 and E 2 over a finite field κ of characteristic p > 0 and any prime l p, the natural map Z l Z Hom(E 1, E 2 ) Hom Zl [Gal(κ/κ)](T l (E 1 ), T l (E 2 )) (where on the left side we consider only homomorphisms defined over κ ) is an isomorphism [114, 1]. The interested reader might find it an instructive exercise to reconstruct this (unpublished) proof by Mumford. Tate proved in [114] that the analogous result holds for all abelian varieties over a finite field and he also incorporated the case l = p by using p-divisible groups. He generalized this result into his influential conjecture [113]: An l-adic cohomology class 2 that is fixed under the Galois group should be a Q l -linear combination of fundamental classes of algebraic cycles when the ground field is finitely generated over its prime field. Honda and Tate gave a classification of isogeny classes of simple abelian varieties A over a finite field κ (see [47] and [117]), and Tate refined this by describing 1 Of course, a supersingular elliptic curve isn t singular. A purist perhaps would like to say an elliptic curve with supersingular j-value. However we will adopt the generally used terminology supersingular elliptic curve instead. 2 The prime number l is assumed to be invertible in the base field. 1

10 2 INTRODUCTION the structure of the endomorphism algebra End 0 (A) (working in the isogeny category over κ) in terms of the Weil q-integer of A, with q = #κ; see [117, Thm. 1]. It follows from Tate s work (see ) that an abelian variety A over a finite field κ admits sufficiently many complex multiplications in the sense that its endomorphism algebra End 0 (A) contains a CM subalgebra 3 L of rank 2 dim(a). We will call such an abelian variety (in any characteristic) a CM abelian variety and the embedding L End 0 (A) a CM structure on A. Grothendieck showed that over any algebraically closed field K, an abelian variety that admits sufficiently many complex multiplications is isogenous to an abelian variety defined over a finite extension of the prime field [86]. This was previously known in characteristic zero (by Shimura and Taniyama), and in that case there is a number field K K such that the abelian variety can be defined over K (in the sense of 1.7.1). However in positive characteristic such abelian varieties can fail to be defined over a finite subfield of K; examples exist in every dimension > 1 (see Example ). Abelian varieties in mixed characteristic. In characteristic zero, an abelian variety A gives a representation of the endomorphism algebra D = End 0 (A) on the Lie algebra Lie(A) of A. If A has complex multiplication by a CM algebra L of degree 2 dim(a) then the isomorphism class of the representation of L on Lie(A) is called the CM type of the CM structure L End 0 (A) on A (see Lemma and Definition ). As we noted above, every abelian variety over a finite field is a CM abelian variety. Thus, it is natural to ask whether every abelian variety over a finite field can be CM lifted to characteristic zero (in various senses that are made precise in 1.8.5). One of the obstacles 4 in this question is that in characteristic zero there is the notion of CM type that is invariant under isogenies, whereas in positive characteristic whatever can be defined in an analogous way is not invariant under isogenies. For this reason we will use the terminology CM type only in characteristic zero. For instance, the action of the endomorphism ring R = End(A 0 ) of an abelian variety A 0 on the Lie algebra of A 0 in characteristic p > 0 defines a representation of R/pR on Lie(A 0 ). Given an isogeny f : A 0 B 0 we get an identification End 0 (A 0 ) = End 0 (B 0 ) of endomorphism algebras, but even if End(A 0 ) = End(B 0 ) under this identification, the representations of this common endomorphism ring on Lie(A 0 ) and Lie(B 0 ) may well be non-isomorphic since Lie(f) may not be an isomorphism. Moreover, if we have a lifting A of A 0 over a local domain of characteristic 0, in general the inclusion End(A) End(A 0 ) is not an equality. If the inclusion End 0 (A) End 0 (A 0 ) is an equality then the character of the representation of End(A 0 ) on Lie(A 0 ) is the reduction of the character of the representation of End(A) on Lie(A). This relation can be viewed as an obstruction to the existence of CM lifting with the full ring of integers of a CM algebra operating on the lift; see 4.1.2, especially , for an illustration. In the case when End(A 0 ) contains the ring of integers O L of a CM algebra L End 0 (A 0 ) with [L : Q] = 2 dim(a 0 ), the representation of O L /po L on Lie(A 0 ) turns out to be quite useful, despite the fact that it is not an isogeny invariant. Its class in a suitable K-group will be called the Lie type of (A 0, O L End(A 0 )). 3 A CM algebra is a finite product of CM fields; see Definition surely also part of the attraction

11 INTRODUCTION 3 The above discrepancy between the theories in characteristic zero and characteristic p > 0 is the basic phenomenon underlying this entire book. Before discussing its content, we recall the following theorem of Honda and Tate ([47, 2, Thm. 1] and [117, Thm. 2]). For an abelian variety A 0 over a finite field κ there is a finite extension κ of κ and an isogeny (A 0 ) κ B 0 such that B 0 admits a CM lifting over a local domain of characteristic zero with residue field κ. Our starting point comes from the following questions. For an abelian variety A 0 over a finite field κ, to ensure the existence of a CM lifting over a local domain with characteristic zero and residue field κ of finite degree over κ, (a) may we choose κ = κ? (b) is an isogeny (A 0 ) κ B 0 necessary? These questions are formulated in various precise forms in 1.8. An isogeny is necessary. Question (b) was answered in 1992 (see [90]) as follows. There exist (many) abelian varieties over F p that do not admit any CM lifting to characteristic zero. The main point of [90] is that a CM liftable abelian variety over F p can be defined over a small finite field. This idea is further pursued in Chapter 3, where the size, or more accurately the minima 5 of the size, of all possible fields of definition of the p-divisible group of a given abelian variety over F p is turned into an obstruction for the existence of a CM lifting to characteristic 0. This is used to show (in 3.8.3) that in most isogeny classes of non-ordinary abelian varieties of dimension 2 over finite fields there is a member that has no CM lift to characteristic 0. (In dimension 1 there are no examples, due to the Deuring Lifting Theorem; see ) We also provide effectively computable examples of abelian varieties over explicit finite fields such that there is no CM lift to characteristic 0. A field extension might be necessary depending on what you ask. Bearing in mind the necessity to modify a given abelian variety over a finite field to guarantee the existence of a CM lifting, we rephrase question (a) in a more precise version (a) below. (a) Given an abelian variety A 0 over a finite field κ of characteristic p, is it necessary to extend scalars to a strictly larger finite field κ κ (depending on A 0 ) to ensure the existence of a κ -rational isogeny (A 0 ) κ B 0 such that B 0 admits a CM lifting over a characteristic 0 local domain R with residue field κ? It turns out there are two quite different answers to question (a), depending on whether one requires the local domain R of characteristic 0 to be normal. The subtle distinction between using normal or general local domains for the lifting went unnoticed for a long time. Once this distinction came in focus, answers to the resulting questions became available. 5 The size of a finite field κ1 is smaller than the size of a finite field κ 2 if κ 1 is isomorphic to a subfield of κ 2, or equivalently if #κ 1 #κ 2. Among the sizes of a family of finite fields there may not be a unique minimal element.

12 4 INTRODUCTION If we ask for a CM lifting over a normal domain up to isogeny, in general a base field extension before modification by an isogeny is necessary. This is explained in 2.1.3, where we formulate the residual reflex obstruction, the idea for which goes as follows. Over an algebraically closed field K of characteristic zero, we know that a simple CM abelian variety B with K-valued CM type Φ (for the action of a CM field L) is defined over a number field in K containing the reflex field E(Φ) of Φ. Suppose that for every K-valued CM type Φ of L, the residue field of E(Φ) at any prime above p is not contained in the finite field κ with which we began in question (a). In such cases, for every CM structure L End 0 (A 0 ) on A 0 and any abelian variety B 0 over κ which is κ-isogenous to A 0, there is no L-linear CM lifting of B 0 over a normal local domain R of characteristic zero with residue field κ. 6 In we give such an example, provided by a supersingular abelian surface over a field with p 2 elements (for any p ±2 (mod 5)). A much broader class of examples is given in 2.3.5, consisting of absolutely simple abelian varieties (with arbitrarily large dimension) over the prime field F p for infinitely many p. Note that passing to the normalization of a complete local noetherian domain generally enlarges the residue field. Hence, if we drop the condition that the mixed characteristic local domain R be normal then the obstruction in the preceding consideration dissolves. And in fact we were put on the right track by mathematics itself. The phenomenon is best illustrated in the example in 4.1.2, which is the same as the example in already mentioned: an abelian surface over a field with p 2 elements which, even up to isogeny, is not CM liftable to a normal local domain of characteristic zero. On the other hand this abelian surface is CM liftable to a mixed characteristic non-normal local domain of characteristic zero. 7 This example is easy to construct, and the proof of the existence of a CM lifting, possibly after applying an F p 2-rational isogeny, is not difficult either. In Chapter 4 we show that the general question of existence of a CM lifting after an appropriate isogeny can be reduced to the same question for (a mild generalization of) the example in 4.1.2, enabling us to prove: every abelian variety A 0 over a finite field κ admits an isogeny A 0 B 0 over κ such that B 0 admits a CM lifting to a mixed characteristic local domain with residue field κ. There are refined lifting problems, such as specifying at the beginning which CM structure on A 0 is to be lifted, or even what its CM type should be on a geometric fiber in characteristic 0. These matters will also be addressed. Our basic method is to localize various CM lifting problems to the corresponding problems for p-divisible groups. Although global properties of abelian varieties are often lost in this localization process, the non-rigid nature of p-divisible groups can be an advantage. In Chapter 3 the size of fields of definition of a p- divisible groups in characteristic p appears as an obstruction to the existence of CM lifting. The reduction steps in Chapter 4 rely on a classification and descent 6 The source of obstructions is that the base field κ might be too small to contain at least one characteristic p residue field of the reflex field E(Φ) for at least one CM type Φ on L. Thus, the field of definition of the generic fiber of the hypothetical lift may be too big. Likewise, an obstruction for question (b) is that the field of definition of the p-divisible group A 0 [p ] may be too big (in a sense that is made precise in 3.8.3). 7 No modification by isogeny is necessary in this example.

13 INTRODUCTION 5 of CM p-divisible groups in characteristic p with the help of their Lie types. In addition, the Serre tensor construction is applied to p-divisible groups, both in characteristic p and in mixed characteristic (0, p); see and for this general construction. Survey of the contents. In Chapter 1 we start with a survey of general facts about CM abelian varieties and their endomorphism algebras. In particular, we discuss the deformation theory of abelian varieties and p-divisible groups, and we review results in Honda-Tate theory that describe isogeny classes and endomorphism algebras of abelian varieties over a finite field in terms of Weil integers. We conclude by formulating various CM lifting questions in 1.8. These are studied in the following chapters. We will see that the questions can be answered with some precision. In Chapter 2 we formulate and study the residual reflex condition. Using this condition we construct several examples of abelian varieties over finite fields κ such that, even after applying a κ-isogeny, there is no CM lifting to a normal local domain with characteristic zero and residue field of finite degree over κ; see 2.3. It is remarkable that many such examples exist, but we do not know whether we have characterized all possible examples; see We then study algebraic Hecke characters and review part of the theory of complex multiplication due to Shimura and Taniyama. Using the relationship between algebraic Hecke characters for a CM field L and CM abelian varieties with CM by L (the precise statement of which we review and prove), we use global methods to show that the residual reflex condition is the only obstruction to the existence of CM lifting up to isogeny over a normal local domain of characteristic zero. We also give another proof by local methods (such as p-adic Hodge theory). In Chapter 3 we take up methods described in [90]. In that paper classical CM theory in characteristic zero was used. Here we use p-divisible groups instead of abelian varieties and show that the size of fields of definition of a p-divisible group in characteristic p is a non-trivial obstruction to the existence of a CM lifting. In 3.3 we study the notion of isogeny for p-divisible groups over a base scheme (including its relation with duality). We show, in one case of the CM lifting problem left open in [90, Question C], that an isogeny is necessary. Our methods also provide effectively computed examples. Some facts about CM p-divisible groups explained in 3.7 are used in 3.8 to get an upper bound of a field of definition for the closed fiber of a CM p-divisible group. In Appendix 3.9, we explain how to use the construction (in 3.7) of a p-divisible group with any given p-adic CM type over the reflex field to produce a semisimple abelian crystalline p-adic representation of the local Galois group such that its restriction to the inertia group is algebraic with algebraic part that we may prescribe arbitrarily in accordance with some necessary conditions (see and 3.9.8). In Chapter 4 we show CM liftability after an isogeny over the finite ground field (with the lifting over a characteristic zero local domain that need not be normal). That is,

14 6 INTRODUCTION every CM structure (A 0, L End 0 (A 0 )) over a finite field κ has an isogeny over κ to a CM structure (B 0, L End 0 (B 0 )) that admits a CM lifting; (see 4.1.1). This statement is immediately reduced to the case when L is a CM field (not just a CM algebra) and the whole ring O L of integers of L operates on A 0, which we assume. Our motivation comes from the proof in (using an algebraization argument at the end of 4.1.3) that the counterexample in to CM lifting over a normal local domain satisfies this property. In general, after an easy reduction to the isotypic case, we apply the Serre-Tate deformation theorem to localize the problem at p-adic places v of the maximal totally real subfield L + of a CM field L End 0 (A 0 ) of degree 2 dim(a 0 ). This reduces the existence of a CM lifting for the abelian variety A 0 to a corresponding problem for the CM p-divisible group A 0 [v ] attached to v. 8 We formulate several properties of v with respect to the CM field L such that any one of them ensures the existence of a CM lifting of A 0 [v ] κ after first applying a κ-isogeny to A 0 [v ] (see 4.1.6, 4.1.7, and 4.5.7). These properties involve the ramification and residue fields of L and L + relative to v. If v violates all of these properties then we call it bad (with respect to L/L + and κ). Let L v := L L + L + v. After applying a preliminary κ-isogeny to arrange that O L End(A 0 ), for v that are not bad we apply an O L -linear κ-isogeny to arrange that the Lie type of the O L,v -factor of Lie(A 0 ) (i.e., its class in a certain K-group of (O L,v /(p)) κ-modules) is self-dual. Under the self-duality condition (defined in 4.4.3) we produce an O L,v -linear CM lifting of A 0 [v ] κ by specializing a suitable O L,v -linear CM v- divisible group in mixed characteristic; see It follows via an argument with deformation rings that if every p-adic place v of L + is not bad then there exists a κ-isogeny A 0 B 0 such that O L End(B 0 ) and the pair (B 0, O L End(B 0 )) admits a lift to characteristic 0 without increasing κ. If the p-adic place v of the totally real field L + is bad then the above argument does not work because in that case no member of the O L,v -linear κ-isogeny class of the p-divisible group A 0 [v ] has a self-dual Lie type. Instead we change A 0 [v ] by a suitable O L,v -linear κ-isogeny so that its Lie type becomes as symmetric as possible, a condition whose precise formulation is called striped. Such a p-divisible group is shown to be isomorphic to the Serre tensor construction applied to a special class of 2-dimensional p-divisible groups of height 4 that are similar to the ones arising from the abelian surface counterexamples in 2.3.1; we call these toy models (see 4.1.3, especially ). These toy models are sufficiently special that we can analyze their CM lifting properties directly; see and (iii). After this key step we deduce the existence of a CM lifting of A 0 [v ] κ from corresponding statements for (the p-divisible group version of) toy models. In the final step we use deformation theory to produce an abelian variety B 0 isogenous to (the original) A 0 over κ and a CM lifting of B 0 over a possibly non-normal 1-dimensional complete local noetherian domain of characteristic 0 with residue field κ. Although O L acts on the closed fiber, we can only ensure that a subring of O L of finite index 9 acts on the lifted abelian scheme (see 4.6.4). 8 See for the statement of the Serre Tate deformation theorem, and and for a precise statement of the algebraization criterion that is used in this localization step. 9 This subring of finite index can be taken to be Z + pol.

15 REFERENCES 7 Appendix A. In Appendix A.1 we provide a self-contained development of the proof of the p-part of Tate s isogeny theorem for abelian varieties over finite fields of characteristic p, as well as a proof of Tate s formula for the local invariants at p-adic places for endomorphism algebras of simple abelian varieties over such fields. (An exposition of these results is also given in [76]; our treatment uses less input from non-commutative algebra.) Appendices A.2 and A.3 provide purely algebraic proofs of the Main Theorem of Complex Multiplication for abelian varieties, as well as a converse result, both of which are used in essential ways in Chapter 2. In Appendix A.4 we use Shimura s method to show that an algebraic Hecke character with a given algebraic part can be constructed over the field of moduli of the algebraic part, with control over places of bad reduction. In the special case of the reflex norm of a CM type (L, Φ), combining this construction of algebraic Hecke characters with the converse to the Main Theorem of CM in A.3 proves that over the associated field of moduli M Q (a subfield of the Hilbert class field of the reflex field E(L, Φ)) there exists a CM abelian variety A with CM type (L, Φ) such that A has good reduction at all p-adic places of M; see A Since M is the smallest possible field of definition given (L, Φ), this existence result is optimal in terms of its field of definition. Typically M E(L, Φ), and this is regarded as a class group obstruction to finding A with its CM structure by L over E(L, Φ), a well-known phenomenon in the class CM theory of elliptic curves. (In the local setting of CM p-divisible groups over p-adic integer rings there are no class group problems and one gets a better result: in 3.7 we use the preceding global construction over the field of moduli to prove that for any p-adic CM type (F, Φ) and the associated p-adic reflex field E Q p over Q p there exists a CM p-divisible group over O E with p-adic CM type (F, Φ).) Appendix B. In Appendices B.1 and B.2, we give two versions of a more direct (but more complicated) proof of the existence of CM liftings for a higherdimensional generalization of the toy model. 10 The first version uses Raynaud s theory of group schemes of type (p,..., p). The second version uses recent developments in p-adic Hodge theory. We hope that material described there will be useful in the future. In Appendix B.3 we compare several Dieudonné theories over a perfect base field of characteristic p > 0. In Appendix B.4 we give a formula for the Dieudonné module of the closed fiber of a finite flat commutative group scheme, constructed using integral p-adic Hodge theory; this formula is used in B.2. References (1) Abelian varieties. In Mumford s book [79] the theory of abelian varieties is developed over an algebraically closed base field, and we need the theory over a general field; references addressing this extra generality are Milne s article [73] (which rests on [79]) and the forthcoming book [42]. Since [42] is not yet in final form we do not refer to it in the main text, but the reader 10 In the original proof of our main CM lifting result in 4.1.1, the case of a bad place v p of L + was reduced through the Serre tensor construction to this existence result. Both B.1 and B.2 are logically independent of results in Chapter 4. Readers who cannot wait to see a proof of the existence of a CM lifting (without modification by any isogeny) for such a higher-dimensional toy model may proceed directly to B.1 or B.2, after consulting 4.2 for the definition of the Lie type of an O-linear p-divisible group and related notation.

16 8 INTRODUCTION should keep in mind that many results for which we refer to [79] and [73] are also treated in [42]. We refer the reader to [80, Ch. 6, 1 2] for a selfcontained development of the elementary properties of abelian schemes, which we freely use. (For example, the group law is necessarily commutative and is determined by the identity section, as in the theory over a field.) (2) Semisimple algebras. We assume familiarity with the classical theory of finite-dimensional semisimple algebras over fields (including the theory of their splitting fields and maximal commutative subfields). A suitable reference for this material is [50, ]; another reference is [10]. In we review some of the facts that we need from that theory. (3) Dieudonné theory and p-divisible groups. To handle p-torsion phenomena in characteristic p > 0 we use Dieudonné theory and p-divisible groups. Brief surveys of some basic definitions and properties in this direction are given in 1.4, , and B B We refer the reader to [115], [68] and [106, 6] for more systematic discussions of basic facts concerning p-divisible groups, and to [28] and [38, Ch. II III] for self-contained developments of (contravariant) Dieudonné theory, with applications to p-divisible groups. Contravariant Dieudonné theory is used in Chapters 1 4. Covariant Dieudonné theory is used in Appendix B.1 because the alternative proof there of the main result of Chapter 4 uses a covariant version of p-adic Hodge theory. A brief summary of covariant Dieudonné can be found in B B We recommend [131] for Cartier theory; an older standard reference is [66]. Notation and terminology Convention on notation. p denotes a prime number. CM fields are usually denoted by L. K often stands for an arbitrary field, κ is usually used to denote either a residue field or a finite field of characteristic p. V denotes the dual of a finite-dimensional vector space V over a field. k denotes a perfect field, often of characteristic p > 0. In , k is an algebraically closed field of characteristic p. K 0 is the fraction field of W (k), where k is a perfect field of characteristic p > 0 and W (k) is the ring of p-adic Witt vectors with entries in k. Abelian varieties are usually written as A, B, or C, and p-divisible groups are often denoted as G or as X or Y. The p-divisible group attached to an abelian variety or an abelian scheme A is denoted by A[p ]; A[p n ] is the subgroup scheme of p n -torsion points in A. Fields and their extensions. For a field K, we write K to denote an algebraic closure and K s to denote a separable closure. An extension of fields K /K is primary if K is separably algebraically closed in K (i.e., the algebraic closure of K in K is purely inseparable over K). For a number field L we write O L to denote its ring of integers. Similar notation is used for non-archimedean local fields.

17 NOTATION AND TERMINOLOGY 9 If q is a power of a prime p, F q denotes a finite field with size q (sometimes understood to be the unique subfield of order q in a fixed algebraically closed field of characteristic p). If κ and κ are abstract finite fields with respective sizes q = p n and q = p n for integers n, n 1 then κ κ denotes the unique subfield of either κ or κ with size p gcd(n,n ) ; the context will always make clear if this is being considered as a subfield of either κ or κ. Likewise, κκ denotes κ κ κ κ, a common extension of κ and κ with size p lcm(n,n ). Base change. If T S is a map of schemes and S is an S-scheme, then T S denotes the S -scheme T S S if S is understood from context. When S = Spec(R) and S = Spec(R ) are affine, we may write T R to denote T R R when R is understood from context. Abelian varieties and homomorphisms between them. The dual of an abelian variety A is denoted A t. For any abelian varieties A and B over a field K, Hom(A, B) denotes the group of homomorphisms A B over K, and Hom 0 (A, B) denotes Q Z Hom(A, B). (Since Hom(A, B) Hom(A K, B K ) is injective, Hom(A, B) is a finite free Z-module since the same holds over K by [79, 19, Thm. 3].) When B = A we write End(A) and End 0 (A) respectively, and call End 0 (A) the endomorphism algebra of A (over K). The endomorphism algebra End 0 (A) is an invariant which only depends on A up to isogeny over K, in contrast with the endomorphism ring End(A). We write A B to denote that abelian varieties A and B over K are K-isogenous. To avoid any possible confusion with notation found in the literature, we emphasize that what we call Hom(A, B) and Hom 0 (A, B) are sometimes denoted by others as Hom K (A, B) and Hom 0 K(A, B). 11 Adeles and local fields. We write A L to denote the adele ring of a number field L, A L,f to denote the factor ring of finite adeles, and A and A f in the case L = Q. If v is a place of a number field L then L v denotes the completion of L with respect to v; O L,v denotes the valuation ring O Lv of L v in case v is non-archimedean, with residue field κ v whose size is denoted q v. For a place w of Q we define L w := Q w Q L = v w L v, and in case w is the l-adic place for a prime l we define O L,l := Z l Z O L = v l O L,v. Class field theory and reciprocity laws. 11 with the notation Hom(A, B) and Hom 0 (A, B) then reserved to mean the analogues for A K and B K over K, or equivalently for A Ks and B Ks over K s.

18 10 INTRODUCTION The Artin maps of local and global class field theory are taken with the arithmetic normalization, which is to say that local uniformizers are carried to arithmetic Frobenius elements. 12 rec L : A L /L Gal(L ab /L) denotes the arithmetically normalized global reciprocity map for a number field L. The composition A L A L /L rec Gal(L ab /L) is denoted r L. For a non-archimedean local field F we write r F : F Gal(F ab /F ) to denote the arithmetically normalized local reciprocity map. Frobenius and Verschiebung. For a commutative group scheme N over an F p -scheme S, N (p) denotes the base change of N by the absolute Frobenius endomorphism of S. The relative Frobenius homomorphism is denoted Fr N/S : N N (p), while the Verschiebung homomorphism for S-flat N of finite presentation denoted Ver N/S : N (p) N (see [29, VII A, ]). If S is understood from context then we may denote these maps as Fr N and Ver N respectively. Likewise, for n 1, the p n -fold relative Frobenius and Verschiebung homomorphisms N N (pn) and N (pn) N are respectively denoted Fr N/S,p n and Ver N/S,p n. Let k be a perfect field of characteristic p > 0, and let σ : W (k) W (k) be the canonical lifting of the Frobenius automorphism y y p of k. A module over the Dieudonné ring D k is a W (k)-module M together with two additive endomorphisms F : M M and V : M M such that F V = [p] M = V F, F(c m) = σ(c) F(m), and c V(m) = V(σ(c) m) for all c W (k) and m M. The semilinear operators F and V on a Dieudonné module M correspond to respective W (k)-linear maps M (p) M and M M (p), where M (p) := W (k) σ,w (k) M. 12 Recall that for a non-archimedean local field F with residue field of size q, an element of Gal(F s/f ) is called an arithmetic (resp. geometric) Frobenius element if its effect on the residue field of F s is the automorphism x x q (resp. x x 1/q ); this automorphism of the residue field is likewise called the arithmetic (resp. geometric) Frobenius automorphism. We choose the arithmetic normalization of class field theory so that uniformizers correspond to Frobenius endomorphisms of abelian varieties in the Main Theorem of Complex Multiplication.

19 CHAPTER 1 Algebraic theory of complex multiplication The theory of complex multiplication... is not only the most beautiful part of mathematics but also of all science. David Hilbert 1.1. Introduction Lifting questions. A natural question early in the theory of abelian varieties is whether every abelian variety in positive characteristic admits a lift to characteristic 0. That is, given an abelian variety A 0 over a field κ with char(κ) > 0, does there exist a local domain R of characteristic zero with residue field κ and an abelian scheme A over R whose special fiber A κ is isomorphic to A 0? We may also wish to demand that a specified polarization of A 0 or subring of the endomorphism algebra of A 0 (or both) also lifts to A. (The functor A A κ from abelian R-schemes to abelian varieties over κ is faithful, by consideration of finite étale torsion levels; see the beginning of ) Suppose there is an affirmative solution to such a lifting problem over some local R. Consider the directed system of noetherian local subrings R i with local inclusion R i R. Since R = inj lim R i, we can descend the solution to such an R i0 except that the residue field κ i0 is merely a subfield of κ. By [32, 0 III, ], there is a faithfully flat local extension R i0 R with R noetherian and having residue field κ over κ i0. By faithful flatness, every minimal prime of R has residue characteristic 0, so we can replace R with its quotient by such a prime to arrive at a solution over a complete local noetherian domain with residue field κ. Thus, an affirmative solution to a lifting problem as above (for a given A 0 ) often amounts to an appropriate deformation ring R for A 0 (over a Cohen ring for κ) admitting a generic point in characteristic 0; the coordinate ring of the corresponding irreducible component of Spec(R) is such an R. If κ κ is an extension of fields and W W is the associated extension of Cohen rings then there is a natural isomorphism R W W R relating the corresponding deformation rings for A 0 and (A 0 ) κ (see and ). Thus, if R has a generic point of characteristic 0 then so does R. Hence, to prove an affirmative answer to lifting questions as above it is usually enough to consider algebraically closed κ. For example, the general lifting problem for polarized abelian varieties (allowing polarizations for which the associated symmetric isogeny A 0 A t 0 is not separable) was solved affirmatively by Norman-Oort [82, Cor. 3.2] when κ is algebraically closed, and the general case then follows by deformation theory (via ). 11

20 12 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION Refinements. When a lifting problem as above has an affirmative solution, it is natural to ask if the (complete local noetherian) base ring R for the lifting can be chosen to satisfy nice ring-theoretic properties, such as being normal or a discrete valuation ring. Slicing methods allow one to find an R with dim(r) = 1 (see for this argument), but normalization generally increases the residue field. Hence, asking that the complete local noetherian domain R be normal or a discrete valuation ring with a specified residue field κ is a non-trivial condition unless κ is algebraically closed. We are interested in versions of the lifting problem for finite κ when we lift not only the abelian variety but also a large commutative subring of its endomorphism algebra. To avoid counterexamples it is sometimes necessary to weaken the lifting problem by permitting the initial abelian variety A 0 to be replaced with another in the same isogeny class over κ. In 1.8 we will precisely formulate several such lifting problems involving complex multiplication, and the main result of our work is a rather satisfactory solution to these lifting problems Purpose of this chapter. Much of the literature on complex multiplication involves either (i) working over an algebraically closed ground field, (ii) making unspecified finite extensions of the ground field, or (iii) restricting attention to simple abelian varieties. To avoid any uncertainty about the degree of generality in which various foundational results in the theory are valid, as well as to provide a convenient reference for subsequent considerations, in this chapter we provide an extensive review of the algebraic theory of complex multiplication over a general base field. This includes special features of the theory over finite fields and over fields of characteristic 0, and for some important proofs we refer to the original literature (e.g., papers of Tate). Some arithmetic aspects (such as reflex fields and the Main Theorem of Complex Multiplication) are discussed in Chapter 2, and Appendix A provides proofs of the Main Theorem of Complex Multiplication and some results of Tate over finite fields Simplicity, isotypicity, and endomorphism algebras Simple abelian varieties. An abelian variety A over a field K is simple (over K) if it is non-zero and contains no non-zero proper abelian subvarieties. Simplicity is not generally preserved under extension of the base field; see Example for some two-dimensional examples over finite fields and over Q. An abelian variety A over K is absolutely simple (over K) if A K is simple Lemma. If A is absolutely simple over a field K then for any field extension K /K, the abelian variety A K over K is simple. Proof. This is an application of direct limit and specialization arguments, as we now explain. Assume the assertion fails with some K /K, so by replacing K with an algebraic closure we may arrange that K is algebraically closed. Choose a non-zero proper abelian subvariety B A K. By expressing K as a direct limit of finitely generated K-subalgebras, there is a finitely generated K-subalgebra R K and an abelian scheme B Spec(R) that is a closed R-subgroup of A R and descends B A K. The constant positive dimension of the fibers of B Spec(R) is strictly less than dim(a). Since K is algebraically closed we can choose a K-point x of

21 1.2. SIMPLICITY, ISOTYPICITY, AND ENDOMORPHISM ALGEBRAS 13 Spec(R), and the fiber B x is a non-zero proper abelian subvariety of A, contrary to the simplicity of A over K. For a pair of abelian varieties A and B over a field K, Hom 0 (A K, B K ) can be strictly larger than Hom 0 (A, B) for some separable algebraic extension K /K. For example, if E is an elliptic curve over Q then considerations with the tangent line over Q force End 0 (E) = Q, but it can happen that End 0 (E L ) = L for an imaginary quadratic field L (e.g., E : y 2 = x 3 x and L = Q( 1)). Scalar extension from number fields to C or from an imperfect field to its perfect closure are useful techniques in the study of abelian varieties, so there is natural interest in considering ground field extensions that are not separable algebraic (e.g., non-algebraic or purely inseparable). It is an important fact that allowing such general extensions of the base field does not lead to more homomorphisms: Lemma (Chow). Let K /K be an extension of fields that is primary (i.e., K is separably algebraically closed in K ). For abelian varieties A and B over K, the natural map Hom(A, B) Hom(A K, B K ) is bijective. Proof. See [22, Thm. 3.19] for a proof using faithfully flat descent (which is reviewed at the beginning of [22, 3]). An alternative proof is to show that the locally finite type Hom-scheme Hom(A, B) over K is étale. We shall be interested in certain commutative rings acting faithfully on abelian varieties, so we need non-trivial information about the structure of endomorphism algebras of abelian varieties. The study of such rings rests on the following fundamental result Theorem (Poincaré reducibility). Let A be an abelian variety over a field K. For any abelian subvariety B A, there is is abelian subvariety B A such that the multiplication map B B A is an isogeny. In particular, if A 0 then there exist pairwise non-isogenous simple abelian varieties C 1,..., C s over K such that A is isogenous to C ei i for some e i 1. Proof. When K is algebraically closed this result is proved in [79, 19, Thm. 1]. The same method works for perfect K, as explained in [73, Prop. 12.1]. (Perfectness is implicit in the property that the underlying reduced scheme of a finite type K- group is a K-subgroup. For a counterexample over any imperfect field, see [24, Ex. A.3.8].) The general case can be pulled down from the perfect closure via Lemma ; see the proof of [22, Cor. 3.20] for the argument Corollary. For a non-zero abelian variety A over a field K and a primary extension of fields K /K, every abelian subvariety B of A K has the form B K for a unique abelian subvariety B A. Proof. By the Poincaré reducibility theorem, abelian subvarieties of A are precisely the images of maps A A, and similarly for A K. Since scalar extension commutes with the formation of images, the assertion is reduced to the bijectivity of End(A) End(A K ), which follows from Lemma

22 14 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION Since any map between simple abelian varieties over K is either 0 or an isogeny, by general categorical arguments the collection of C i s (up to isogeny) in the Poincaré reducibility theorem is unique up to rearrangement, and the multiplicities e i are also uniquely determined Definition. The C i s in the Poincaré reducibility theorem (considered up to isogeny) are the simple factors of A. By the uniqueness of the simple factors up to isogeny, we deduce: Corollary. Let A be a non-zero abelian variety over a field, with simple factors C 1,..., C s. The non-zero abelian subvarieties of A are generated by the images of maps C i A from the simple factors Central simple algebras. Using notation from the Poincaré reducibility theorem, for a non-zero abelian variety A we have End 0 (A) Mat ei (End 0 (C i )) where {C i } is the set of simple factors of A and the e i s are the corresponding multiplicities. Each End 0 (C i ) is a division algebra, by simplicity of the C i s. Thus, to understand the structure of endomorphism algebras of abelian varieties we need to understand matrix algebras over division algebras, especially those of finite dimension over Q. We therefore next review some general facts about such rings Definition. A central simple algebra over a field K is a non-zero associative K-algebra of finite dimension such that K is the center and the underlying ring is simple (i.e., has no non-trivial two-sided ideals). A central division algebra over K is a central simple algebra over K whose underlying ring is a division algebra. Among the most basic examples of central simple algebras over a field K are the matrix algebras Mat n (K) for n 1. The most general case is given by: Proposition (Wedderburn s Theorem). Every central simple algebra D over a field K is isomorphic to Mat n ( ) = End ( n ) for some n 1 and some central division algebra over K. Moreover, n is uniquely determined by D, and is uniquely determined up to K-isomorphism. Proof. This is a special case of a general structure theorem for simple rings; see [50, Thm. 4.2] and [50, 4.4, Lemma 2]. In addition to matrix algebras, another way to make new central simple algebras from old ones is to use tensor products: Lemma. If D and D are central simple algebras over a field K, then so is D K D. For any extension field K /K, D K := K K D is a central simple K -algebra. Proof. The first part is [50, 4.6, Cor. 3] and the second is [50, 4.6, Cor. 1, 2].

23 1.2. SIMPLICITY, ISOTYPICITY, AND ENDOMORPHISM ALGEBRAS Splitting fields. It is a general fact that for any central division algebra over a field K, Ks is a matrix algebra over K s (so [ : K] is a square). In other words, is split by a finite separable extension of K. There is a refined structure theory concerning splitting fields and maximal commutative subfields of central simple algebras over fields, and we refer the reader to [50, ] for a self-contained development of this material. An important result in this direction is: Proposition. Let D be a central simple algebra over a field F, with [D : F ] = n 2. An extension field F /F with degree n embeds as an F -subalgebra of D if and only if F splits D (i.e., D F Mat n (F )). Moreover, if D is a division algebra then every maximal commutative subfield of D has degree n over F. Proof. The first assertion is a special case of [50, Thm. 4.12]. Now assume that D is a division algebra and consider a maximal commutative subfield. In such cases F splits D (by [50, 4.6, Cor. to Thm. 4.8]), so n [F : F ] by [50, Thm. 4.12]. To establish the reverse divisibility it suffices to show that for any central simple algebra of dimension n 2 over F, every commutative subfield of D has F -degree at most n. If A is any simple F -subalgebra of D and its centralizer in A is denoted Z D (A) then n 2 = [A : F ][Z D (A) : F ] [50, 4.6, Thm. 4.11]. Thus, if A is also commutative (so A Z D (A)) then [A : F ] n. The second assertion in Proposition does not generalize to central simple algebras; e.g., we may have D = Mat n (F ) with F having no degree-n extension fields. We conclude our general discussion of central simple algebras by reviewing the K-linear reduced trace map Trd D/K : D K for a central simple algebra D over a field K Construction. Let D be a central simple algebra over an arbitrary field K. It splits over a separable closure K s, which is to say that there is a K s -algebra isomorphism f : D Ks Mat n (K s ) onto the n n matrix algebra for some n 1. By the Skolem-Noether theorem, all automorphisms of a matrix algebra are given by conjugation by an invertible matrix. Hence, f is well-defined up to composition with an inner automorphism. The matrix trace map Tr : Mat n (K s ) K s is invariant under inner automorphisms and is equivariant for the natural action of Gal(K s /K), so the composition of the matrix trace with f is a K s -linear map D Ks K s that is independent of f and Gal(K s /K)-equivariant. Thus, this descends to a K-linear map Trd D/K : D K that is defined to be the reduced trace. In other words, the reduced trace map is a twisted form of the usual matrix trace, just as D is a twisted form of a matrix algebra. (For d D, the K-linear left multiplication map x d x on D has trace [D : K] Trd D/K (x), as we can see by scalar extension to K s and a direct computation for matrix algebras. The elimination of the coefficient [D : K] is the reason for the word reduced.) Brauer groups. For applications to abelian varieties it is important to classify division algebras of finite dimension over Q (such as the endomorphism algebra of a simple abelian variety over a field). If is such a ring then its center