Grundlehren der Mathematischen Wissenschaften, 255. New York etc.: Springer-Verlag. VIII, 184 p. DM 128.00; {$} 49.70 (1983).

Abelian varieties with many complex multiplications have attracted research for at least two reasons. Firstly, they provide explicit constructions of class fields for a large family of number fields, in a manner initiated, after Hecke, by {\it A. Weil, G. Shimura} and {\it Y. Taniyama} [Proc. Symp. Algebraic Number Theory, Tokyo-Nikko 1955, 9--22, 23--33, 31--45 (1956;

Zbl 0074.26802;

Zbl 0074.26602;

Zbl 0074.26801)] following the classical results of Kronecker, Weber, Tagaki and Deuring for imaginary quadratic fields. Secondly, they prove to be a good testing ground for conjectures on general Abelian varieties, or even on general varieties, as, e.g., in the context of $L$-functions or in the study of their Hodge ring. Until recently, mathematicians interested in their properties had essentially three books at their disposal: the classical monograph of {\it G. Shimura} and {\it Y. Taniyama} [Complex multiplication of Abelian varieties and applications to number theory. Publications of the Mathematical Society of Japan, 6. Tokyo: The Mathematical Society of Japan (1961;

Zbl 0112.03502), quoted [S-T] below], the book of {\it G. Shimura} [Introduction to the arithmetic theory of automorphic functions. Publications of the Mathematical Society of Japan. 11. Tokyo: Iwanami Shoten, Publishers and Princeton, NJ: Princeton University Press (1971;

Zbl 0221.10029), quoted [Sh] below] which, although centering on the elliptic case, gives a lot of information in higher dimensions, and the lecture notes of {\it G. Shimura} [Automorphic functions and number theory. Lecture Notes in Mathematics. Vol. 54, Berlin etc.: Springer (1968;

Zbl 0183.25402)] which describe the results without proofs. Here is a fourth book, whose author does not conceal his debt to the first two works mentioned above. Indeed, the fundamental material of the theory (in general properties of CM types, reduction mod $\wp$, the congruence relations, the main theorem of complex multiplication and the description of fields of moduli), which makes up for an important part of the first five chapters of the book, is organized in close connection with [S-T] or [Sh]. Applications and refinements, of a different flavour, are discussed in the remaining two chapters. We now analyse the contents of each chapter in more detail, and comment on them at the same time.
Chapter I introduces CM types and the analytic theory of the associated Abelian variety, recalling without proofs the standard facts on Abelian manifolds. The case of cyclotomic fields is studied as an application, after Koblitz and Rohrlich. Reduction mod $\wp$, in the terminology of [S-T], and the basic facts on $\ell$-adic representations are discussed in chapter II. In addition, the theorem of Serre and Tate characterizing primes of good reduction is stated with a sketch of proof (which forces the author to turn to scheme terminology -- he could perhaps have done so from the start). The heart of the matter comes in chapter III, with a proof of the congruence relations, giving the prime ideal decomposition of the endomorphism lifting the Frobenius map on the reduced variety. (It seems a pity that no application to the elliptic case, i.e. to Kronecker’s relation, is here given, if only to explain the denomination of this result; in fact, the author seems to have deliberately refrained in this book from such illustrations, apart from a handful of short remarks, as on p. 92 for the zeta function, or on pp. 132 and 138 for explicit generators of class fields.) A clear description of the behaviour of Riemann forms under pull-backs is then given; as elementary though it is, this is a key point in the theory, and the author is right in displaying it. (He does not, however, point out why polarizations restrict the action of the Galois group, although this is immediately apparent in terms of the Weil pairing -- one must wait till p. 172 in chapter VII to get a hint on this point. Such a discussion could have paved the way to the definition of the Hodge group, thereby introducing the reader to an important aspect of current research on the topic.) The chapter closes with the main theorem of complex multiplication: The action on the torsion points of an automorphism of the Abelian closure of the reflex field $K'$ is given by the determinant $N'$ of the reflex type acting on the corresponding idèle. The proof assumes knowledge of class field theory, and follows the program suggested in [Sh], 5.5.C. The rest of the book describes refinements and illustrations of this beautiful result, together with more recent developments of the theory. In chapter IV, the field of definition $k$ of the Abelian variety and its endomorphisms is taken as ground field, as in [Sh], 7.8. The action of the Galois group of the Abelian closure of $k$ is then given by a character on the idèle classes of $k$ whose product $\alpha$ by $N\sb{k/K'}{\Bbb O}N'$ is studied in a clear way under the denomination of “CM character”. Applications of various natures are given, e.g. to the Lang-Manin-Mumford conjecture, or, after Shimura, to the zeta function of the Abelian variety over fields of definition possibly smaller than $k$. Chapter V describes Kummer varieties, which provide a construction of class fields of $K'$, and the possibility of descent to smaller fields -- cf. [S-T], §§ 4 and 16, and [Sh], p. 130. (The fact that the associated fields of moduli do not yield all class fields of $K'$ could have been stated more explicitly.) Chapter VI introduces Kubota’s notion of the rank of a CM type, and describes Ribet’s computation of the image of N’, and therefore of the image of the Galois group in the different $\ell$-adic representations; following Serre, one views $N'$ as a homomorphism of linear tori, obtained by restriction of scalars. Finally, chapter VII presents some results and a conjecture of Tate on type transfers and on the action of the Galois group over an arbitrary ground field, and the solution of this conjecture as a corollary of Deligne’s relations on generalized CM types (as mentioned by the author, this is a special case of Langlands’ conjecture on conjugation of Shimura varieties; a precise reference to the articles of {\it J. S. Milne} and {\it K.-Y. Shih} in Lect. Notes Math. 900, 229--260, 280--356 (1982;

Zbl 0478.12011,

Zbl 0478.14029) would here have been enlightening). As may be apparent from this synopsis, the book is dense, and requires from the reader a good command of the theory of Abelian varieties, and of class field theory. However, little warning is given of this point. In fact, the level of the book is not clearly stated, and is often jumpy: elementary results are sometimes proved (e.g. on commutants, p. 12, on norms of ideals, p. 58), while subtle points (especially in connection with algebraic geometry, cf. p. 47 on Néron models, p. 159 on the dimension of the algebraic group B) are assumed. In this respect, it would have been helpful to gather in an introduction the basic facts needed on Abelian varieties, rather than scatter them along the first three chapters when they are about to be used, and to recall the fundamental exact sequence of class field theory (as, e.g. in [Sh], 5.2). But my main reservation concerns the absence of several important recent aspects of complex multiplication. It seems to me that a book written in 1983 on this subject should have at least mentioned such results as Mumford’s counterexample to Tate’s 1--1 conjecture [cf. {\it A. Weil}, Œ uvres scientifiques, Vol. III (1979;

Zbl 0424.01029), 421--429], Shimura’s relations on periods [{\it G. Shimura}, J. Math. Soc. Japan 31, 561--592 (1979;

Zbl 0456.10015)] or the key role played by Abelian varieties of CM type in Deligne’s study of absolute Hodge cycles, in order to guide the reader into the current advances of the theory. On the other hand, several aspects of the book, especially the lucid description of the CM character, or the study of the size of the Galois group of torsion points, make it worthwhile reading.
Finally, we mention that the typography of the book is very pleasant, with formulae and main results clearly displayed. It is unfortunately marred by a number of misprints (to quote a few: $S$ for $S\sb F$ on p. 8, l. 5; linear instead of algebraic equivalence on p. 68, l. 5; $K'\sb{ab}$ for $K'\sb{ab}\cap k\sp{ab}$ on p. 85, l. 9; $\text{Pic}$ for $\text{Pic}\sb 0$ on p. 129, l. 7), and some of the references of the last chapter ([DPP] on p. 163; [Shih] on p. 171) are erratic.