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. Language: English