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MR0106225 (21 #4959) Generally speaking, this book is very well written and would give investigators an excellent account of what has been done, without going through many papers. Also it is convenient that this book contains all results in Weil's book on Abelian varieties [ Vari\'et\'es ab\'eliennes et courbes alg\'ebriques, Actualités Sci. Ind. no. 1064, Hermann, Paris, 1948; MR 10, 621] (except possibly a construction of a group variety from a variety having a normal law of composition); some proofs are simplified and are made lucid. However, it is regrettable that some of the basic and useful theorems on Abelian varieties (such as duality, absence of torsion, Riemann-Roch theorem, theorem of Frobenius, etc.) had to be omitted (partly because these were not available at the time when the book was being prepared). The following list of chapter headings and comments will make the scope of this book clear. In the following, we assume that all ambient varieties are non-singular in co-dimension 1. Chapter I contains preparatory remarks about group varieties, and some of them have no proofs (references are given). It might have been a good idea, in order to make this book self-contained, to include at least a construction of a group variety from a variety with a normal law of composition. Contents of Chapter II are: properties of rational mappings of varieties into Abelian varieties (which include the Poincaré complete reducibility theorem); construction of Jacobian varieties of curves; construction of Albanese varieties of given varieties. Chapter III starts with the definition of algebraic equivalence of cycles and gives a proof of Weil's theorem of the square: $X(a,b)-X(a,b')-X(a',b)+X(a',b')\sim 0$ on a product $U\times V\times W$. From this, it follows that if $X$ is a divisor on $G\times W$, where $G$ is a group variety, $X(a_1)-X(a_2)-X(a_3)+X(a_4)\sim 0$ if $a_1a_3{}^{-1}=a_2a_4{}^{-1}$, which is a fundamental theorem in the theory of Picard varieties as treated here. After this, the symbol $\varphi_X$ is defined for a divisor $X$ on a commutative group variety $G$, as a homomorphism $a\rightarrow\text{Cl}(X_a-X)$ of $G$ into the group of divisor classes $\text{Pic}\,(G)$ of $G$ with respect to linear equivalence. Then the theorem of the square shows that the kernel of $\varphi_X$ is the algebraic subgroup of $G$. Chapter IV begins by proving an existence of a positive non-degenerate divisor $X$ on an Abelian variety $A^r$ ($X$ is such that the kernel of $\varphi_X$ is finite). From this, the Picard variety of $A$ is constructed. This chapter contains a proof of the theorem that $\nu(\sum m_i\alpha_i)$ is homogeneous of degree $2r$ in the $m_i$ with rational coefficients (in particular $\nu(n\delta)=n^{2r}$), together with related topics, which is simpler than the one given by Weil originally and is based on the author's own contribution. In Chapter V, first the transpose ${}^t\alpha$ of a homomorphism $\alpha$ of an Abelian variety $A$ into another Abelian variety $B$ is defined, then some related formulas are proved. Next, assuming $A=B$, an involution $\alpha\rightarrow\varphi_X{}^{-1}{}^t\alpha\varphi_X=\alpha'$ is given and the Castelnuovo inequality $\text{tr}\,(\alpha\alpha')>0$ (if $\alpha\neq 0$) is proved. From this, it is proved that if $A$ is defined over a finite field $k$ with $q$ elements, absolute values of characteristic roots of the Frobenius endomorphism relative to $k$ are all equal to $\surd q$. The chapter ends with a few remarks about positive endomorphisms. Chapter VI begins with the existence theorem of the Picard variety of a given variety. Then the theorem of divisorial correspondences is given. Using this and results of Chapter V, a proof of the Riemann hypothesis for curves over finite fields is given. Finally, the reciprocity theorem $f((g))=g((f))$ on curves is generalized to Abelian varieties. In Chapter VII, $l$-adic representations of homomorphisms are discussed, more neatly than in Weil's book. As a consequence, the structure of the module of homomorphisms is given, among other things. The chapter ends with a remark about a polarized Abelian variety. In Chapter VIII, Chow's theory of $K/k$-image and $K/k$-trace of an Abelian variety defined over $K$ is given. It also contains results about exact sequences, which are sometimes useful. Finally, an Appendix is added to include some auxiliary results about correspondences used in this book. Readers will find an "Historical Note" at the end of each chapter, about works of algebraic geometers, mainly those which were done after Weil's Foundations of algebraic geometry\/ [Amer. Math. Soc., New York, 1946; MR 9, 303] had appeared. This might be helpful sometimes. Also some problems are mentioned. \{Remarks: In the Note to Chapter VI, the theory of correspondences, as formulated here, is attributed to Severi, but at least Hurwitz's name should have been mentioned also, since the theory of correspondences itself is essentially due to him (expressed in terms of Abelian integrals). Also, elsewhere, a remark is made that the seesaw principle is implicit in Severi's work, but as far as the reviewer knows, it was rather explicit.\} Reviewed by T. Matsusaka |