Complex Multiplication J.S. Milne July 14, 2020 These are preliminary notes for a modern account of the theory of complex multipli- cation. The article The fundamental theorem of complex multiplication (2007) is a concise version of part of this work. Probably the two should be read together. v0.00 April 7, 2006. First version posted on the web; 113 pages. v0.10 July 14, 2020. Updated TeX; minor fixes; 108 pages. None Please send comments and corrections to me at jmilne at umich dot edu. Available at https://www.jmilne.org/math/CourseNotes/ Copyright c 2006, 2020. J.S. Milne. Contents Contents3 I Analytic Theory8 1 CM-algebras and CM-types............................8 Review of semisimple algebras and their modules 8; CM-algebras 10; CM-types. 11; The reflex field of a CM-pair 13; The reflex norm. 14; Classification of the primitive CM-pairs 18; Positive involutions and CM-algebras 19 2 Complex abelian varieties.............................21 Complex tori 21; The cohomology of complex tori 23; Hermitian forms and alternating forms 24; Riemann forms 25; Abelian varieties 26 3 Abelian varieties with complex multiplication..................27 Definition of CM abelian varieties 27; The reflex field of an abelian variety with complex multiplication 29; Classification up to isogeny 29; Classification up to isomorphism 31 4 Mumford-Tate groups...............................31 Review of algebraic groups of multiplicative type 31; CM-pairs and tori 34; The reflex norm in terms of tori 35; Complex multiplication in terms of tori 35; Mumford-Tate groups 35; Infinity types 37; The Serre group 40 ; Abelian varieties of CM-type 44 5 Motives......................................45 The Hodge structure of an abelian variety 45; Abelian motives 45; Hodge structures 45; CM-motives 45 II The Arithmetic Theory 46 6 Abelian varieties and their good reductions ...................46 Complex abelian varieties and complex tori 46; Specialization of abelian varieties 48; The good reduction of abelian varieties 48 7 Abelian varieties with complex multiplication..................52 Definition of a CM abelian variety 52; Complex multiplication by a Q-algebra 52; Specialization 54; Rigidity 54; Good reduction 55; The degrees of isogenies 55 ; a-multiplications: first approach 58; a-multiplications: second approach 62; a-multiplications: complements 64 8 The Shimura-Taniyama formula .........................66 Review of numerical norms 66; Statement and proof 66; Alternative approach using schemes (Giraud 1968) 70; Alternative approach using p-divisible groups (Tate 1968) 71 ; Alternative approach using crystals (Deligne c1968) 73; Alternative approach using Hodge-Tate decompositions (Serre 1968) 73 9 The fundamental theorem over the reflex field..................74 Review of the reflex norm 74; Preliminaries from algebraic number theory 74; The funda- mental theorem in terms of ideals 75; More preliminaries from algebraic number theory 3 CONTENTS 4 76; The fundamental theorem in terms of ideles` 77; The fundamental theorem in terms of uniformizations 80; The fundamental theorem in terms of moduli 81 ; Alternative approach using crystals (Deligne c1968) 83 10 The fundamental theorem of complex multiplication ..............88 Statement of the Theorem 88; Definition of f˚ ./ 90; Proof of Theorem 10.2 up to an element of order 2 93; Completion of the proof (following Deligne) 95 III CM-Motives 97 IV Applications 98 A Additional Notes; Solutions to the Exercises 99 B Summary 101 Bibliography 104 Index of definitions 107 Preface Abelian varieties with complex multiplication1 are special in that they have the largest pos- sible endomorphism rings. For example, the endomorphism ring of an elliptic curve is usually Z, but when it is not, it is an order in an imaginary quadratic number field, and the elliptic curve is then said to have complex multiplication. Similarly, the endomorphism ring of a simple abelian variety of dimension g is usually Z, but, at the opposite extreme, it may be an order in a number field of degree 2g, in which case the abelian variety is said to have complex multiplication. Abelian varieties with complex multiplication correspond to special points on the moduli variety of abelian varieties, and their arithmetic is intimately related to that of the values of modular functions and modular forms at those points. The first important result in the subject, which goes back to Kronecker and Weber, states that the Hilbert class field (maximal abelian unramified extension) of an imaginary quadratic subfield E of C is generated by the special value j./ of the j -function at any element of E in the complex upper half plane generating the ring of integers in E. Here j is the holomorphic function on the complex upper half plane invariant under the action 1 p 3 of SL2.Z/, taking the values 0 and 1728 respectively at C and p 1, and having a 2 simple pole at infinity. The statement is related to elliptic curves through the ideal class group of E, which acts naturally both on the Hilbert class field of E and on the set of isomorphism classes of elliptic curves with endomorphism ring OE . Generalizing this, Hilbert asked in the twelfth of his famous problems whether there ex- ist holomorphic functions whose special values generate the abelian extensions (in particu- lar, the class fields) of arbitrary number fields. For imaginary quadratic fields, the theory of elliptic curves with complex multiplication shows that elliptic modular functions have this property (Kronecker, Weber, Takagi, Hasse). Hecke began the study of abelian surfaces with complex multiplication in the early 1900s, but the primitive state of algebraic geome- try over fields other than C made this premature. It was not until the 1950s, after Weil had developed the theory of abelian varieties in arbitrary characteristic, that he, Shimura, and Taniyama were able to successfully extend the main statements of the theory of complex multiplication from elliptic curves to abelian varieties. While the resulting theory has pro- vided only a partial answer to Hilbert’s problem, it has played an essential role in the theory of modular (and, more generally, Shimura) varieties and in other aspects of number theory. The complex points of a modular variety parametrize polarized abelian varieties over C together with a level structure; at a special point, the abelian variety has complex multi- plication. To understand the arithmetic nature of the values of modular functions at these special points, it is necessary to understand how abelian varieties with complex multiplica- 1The name is both archaic and imprecise — the term “multiplication” is no longer used to denote an endomorphism, and “complex multiplication” is sometimes used to denote a more general class (Birkenhake and Lange 2004, p. 262) — but I know of no other. 5 CONTENTS 6 tion and their torsion points behave under automorphisms of C (as an abstract field). For automorphisms of C fixing a certain “reflex” field attached to the abelian variety, this is the main content of the theory of Shimura, Taniyama, and Weil from the 1950s. Their results were extended to all automorphisms of C by the later work of Deligne, Langlands, and Tate. NOTATION. By a field we always mean a commutative field. A number field is a field of finite degree over Q.2 An algebraic closure of a field k is denoted kal. We let C denote an algebraic closure of and al the algebraic closure of in . We often use to denote an algebraic R Q Q C QN closure of Q (not necessarily Qal). Complex conjugation on C (or a subfield) is denoted by or simply by a a.A complex conjugation on a field k is an involution induced by 7! N complex conjugation on C and an embedding of k into C.3 An automorphism of a field ˝ is said to fix a subfield k if a a for all a k. D 2 When k is a field, an etale´ algebra over k is a finite product of finite separable field extensions of k. Let E be an etale´ Q-algebra, and let k be a field containing Q. We say that k contains all conjugates of E if every Q-algebra homomorphism E kal maps into k; ! equivalently, if there are ŒE Q distinct Q-algebra homomorphisms E k. W ! Rings are assumed to have a 1, homomorphisms of rings are required to map 1 to 1, and 1 acts on every module as the identity map. By a k-algebra (k a field) I mean a ring B containing k in its centre. Following Bourbaki, I require compact topological spaces to be separated (Hausdorff). An algebraic variety over a field k is a geometrically reduced scheme of finite type over k. If V and V are algebraic varieties over a field k, then a morphism V V means 0 ! 0 a morphism (regular map) defined over k. If K is a field containing k, then VK is the algebraic variety over K obtained by extension of the base field and V .K/ is the set of points of V with coordinates in K.4 If k , K is a homomorphism of fields and V is an W ! algebraic variety (or other algebro-geometric object) over k, then V has its only possible meaning: apply to the coefficients of the equations defining V . The tangent space at a point P of a variety V is denoted by TgtP .V /. Let A and B be sets and let be an equivalence relation on A. If there exists a canonical surjection A B whose fibres are the equivalence classes, then I say that B classifies the ! elements of A modulo or that it classifies the -classes of elements of A. REFERENCES In addition to those listed at the end, I refer to the following of my books and course notes (the latter are available at www.jmilne.org/math/).