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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 Chapter 6 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 . . . . .

Inhaltsverzeichnis

Frontmatter

Chapter 1. Analytic Complex Multiplication

Abstract
This chapter is essentially elementary, and lays the foundations for the study of the endomorphisms of complex toruses known as complex multiplications. Let V be a vector space of dimension n over the complex numbers. Let Λ be a lattice in V. The quotient complex analytic group V/Λ is called a complex torus. We assume known the basic facts concerning Riemann forms and the projective embedding of such toruses. A (non-degenerate) Riemann form E on V/Λ is an alternating non-degenerate form on V such that E(x, y) ∊ Z for x, y ∊ Λ, and such that the form E(ix, y) is symmetric positive definite. Equivalently, one may say that E is the imaginary part of a positive definite hermitian form on V, and takes integral values on Λ. The torus admits a projective embedding if and only if it admits a Riemann form, and such a projective embedding is obtained by projective coordinates given by theta functions. We shall not need to know anything about such theta functions aside from their existence. An abelian manifold is a complex torus which admits a Riemann form.
Serge Lang

Chapter 2. Some Algebraic Properties of Abelian Varieties

Abstract
Of necessity, this chapter duplicates to a large extent the corresponding material in Shimura-Taniyama [Sh-T]. We make the transition to the algebraic situation, and we assume that the reader is acquainted with the general theory of abelian varieties. No matter whether the reader picks up the basic properties from my book on the subject (after Weil), Mumford, Shimura (for reduction mod p), or the foundations laid by the Grothendieck school, the reader should end up knowing the same basic theorems. Because of my background, I use the terminology of Weil (generic points when needed), and the language of reduction mod p is that of Shimura. I have recalled with proofs some elementary definitions and properties, and without proof some of the more advanced results in this direction.
Serge Lang

Chapter 3. Algebraic Complex Multiplication

Abstract
This chapter contains the first fundamental theory of complex multiplication. When an abelian variety has a sufficiently large ring of endomorphisms, then the Frobenius endomorphism of the variety mod p can be represented as the reduction mod p of an element in that ring, which is, say, the ring of integers in a number field K. If π is that element, then a basic theorem gives the ideal factorization of π in DK. We have followed Shimura-Taniyama for the proof of this result. On the other hand, Shimura in his book [Sh 1] gave a formulation in terms of ideles, and suggested that one could give a proof for this more general form, directly from the factorization theorem. We have carried out this approach.
Serge Lang

Chapter 4. The CM Character

Abstract
The first main theorem dealt with the reflex field K’ as ground field. We shall now deal with the field of definition k itself as ground field. Then we shall see that k(Ator) is abelian over k, and we shall obtain an abelian character out of the situation. By definition, a character is a continuous homomorphism.
Serge Lang

Chapter 5. Fields of Moduli, Kummer Varieties, and Descents

Abstract
This chapter deals with the analysis of fields of definition for the various structures we have encountered.
Serge Lang

Chapter 6. The Type Norm

Abstract
In Chapter 4, Theorem 2.8, we reduced the study of certain subgroups of finite index in Galois groups of torsion points to the image of the type norm
$${{N}_{{{{{\Phi '}}_{k}}}}}:\mathfrak{o}_{{k'}}^{*} \to \mathfrak{o}_{{K,p}}^{*}. $$
.
Serge Lang

Chapter 7. Arbitrary Conjugations of CM Types

Abstract
This chapter is based on an unpublished article of Tate [Ta], who formulated a conjecture extending the fundamental theorem of complex multiplication to the case when the automorphism σ does not leave the reflex field fixed. Tate obtains a commutative diagram just as before, up to an idele of square 1, thus leaving the conjecture that this idele can in fact be taken to be 1. This conjecture is equivalent to an important special case of a conjecture of Langlands concerning the conjugation of Shimura varieties [Lglds]. Tate reformulates the conjecture in terms of a “type transfer”. The first two sections of the chapter give the general algebraic number theory setting for this type transfer, and the final sections give the application to the abelian varieties with complex multiplication.
Serge Lang

Backmatter

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