Complex Multiplication

The small book by Shimura-Taniyama on the subject of complex multi­ is a classic. It gives the results obtained by them (and some by Weil) plication in the higher dimensional case, generalizing in a non-trivial way the method of Deuring for elliptic curves, by reduction mod p. Partly through the work of Shimura himself (cf. [Sh 1] [Sh 2], and [Sh 5]), and some others (Serre, Tate, Kubota, Ribet, Deligne etc.) it is possible today to make a more snappy and extensive presentation of the fundamental results than was possible in 1961. Several persons have found my lecture notes on this subject useful to them, and so I have decided to publish this short book to make them more widely available. Readers acquainted with the standard theory of abelian varieties, and who wish to get rapidly an idea of the fundamental facts of complex multi­ plication, are advised to look first at the two main theorems, Chapter 3, §6 and Chapter 4, §1, as well as the rest of Chapter 4. The applications of Chapter6 could also be profitably read early. I am much indebted to N. Schappacher for a careful reading of the manu­ script resulting in a number of useful suggestions. S. LANG Contents CHAPTER 1 Analytic Complex Multiplication 4 I. Positive Definite Involutions . . . 6 2. CM Types and Subfields. . . . . 8 3. Application to Abelian Manifolds. 4. Construction of Abelian Manifolds with CM 14 21 5. Reflex of a CM Type . . . . .

1 Analytic Complex Multiplication.- 1. Positive Definite Involutions.- 2. CM Types and Subfields.- 3. Application to Abelian Manifolds.- 4. Construction of Abelian Manifolds with CM.- 5. Reflex of a CM Type.- 6. Application to Cyclotomic Fields.- 7. An Example: The Fermat Curve.- 2 Some Algebraic Properties of Abelian Varieties.- 1. Invariant Differential Forms.- 2. Homomorphisms and Inseparability.- 3. Reduction mod p and l-adic Representations.- 4. Reduction of Functions.- 5. Reduction of Differential Forms.- 3 Algebraic Complex Multiplication.- 1. Fields of Definition.- 2. Transformations and Multiplications.- 3. The Congruence Relation.- 4. Polarizations.- 5. Change of Riemann Forms Under Various Maps.- 6. The Main Theorem of Complex Multiplication.- 4 The CM Character.- 1. The Second Main Theorem of Complex Multiplication and the CM Character.- 2. Finite Extensions.- 3. Algebraic Properties of the Associated Characters.- 4. The CM Character over a Quadratic Subfield.- 5. Shimura’sl-adic Representations.- 6. Application to the Zeta Function in the CM Case.- 5 Fields of Moduli, Kummer Varieties, and Descents.- 1. Fields of Moduli.- 2. General Descent.- 3. Kummer Varieties.- 4. Class Fields as Moduli Fields.- 5. Casselman’s Theorem.- 6. Descent to a Quadratic Subfield.- 7. Further Descent Theorems.- 6 The Type Norm.- 1. The Rank of a Type.- 2. The Type Norm as Lie Homomorphism.- 3. The Image N?(Dp?).- 4. The Type Norm as Algebraic Homomorphism.- 5. Application to Abelian Varieties.- 7 Arbitrary Conjugations of CM Types.- 1. The Reflex Norm and the Type Transfer.- 2. General Reciprocity and the Type Transfer.- 3. Application to the Conjugation of Abelian Varieties.- 4. Another Property Implying e? =1.