Complex Multiplication

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’s l-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.