Based on the work in algebraic geometry by Norwegian mathematician Niels Henrik Abel (1802–29), this monograph was originally published in 1959 and reprinted later in author Serge Lang's career without revision. The treatment remains a basic advanced text in its field, suitable for advanced undergraduates and graduate students in mathematics. Prerequisites include some background in elementary qualitative algebraic geometry and the elementary theory of algebraic groups.
The book focuses exclusively on Abelian varieties rather than the broader field of algebraic groups; therefore, the first chapter presents all the general results on algebraic groups relevant to this treatment. Each chapter begins with a brief introduction and concludes with a historical and bibliographical note. Topics include general theorems on Abelian varieties, the theorem of the square, divisor classes on an Abelian variety, functorial formulas, the Picard variety of an arbitrary variety, the I-adic representations, and algebraic systems of Abelian varieties. The text concludes with a helpful Appendix covering the composition of correspondences.
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About the Author
Serge Lang (1927–2005) graduated from the California Institute of Technology and earned his Ph.D. in mathematics at Princeton in 1951. He taught at the University of Chicago, Columbia, and at Yale from 1972 until the end of his career. His research focused on geometry and number theory, but he taught and wrote books about many areas in mathematics, becoming one of the field's most prolific authors of advanced monographs.
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Pour des simplifications plus substantielles, le développement futur de la géometrie algebrique ne saurait manquer sans doute d'en faire apparaitre.
It is with considerable pleasure that we have seen in recent years the simplification s expected by Weil realize themselves, and it has seemed timely to incorporate them into a new book.
We treat exclusively abelian varieties, and do not pretend to write a treatise on algebraic groups. Hence we have summarized in a first chapter all the general results on algebraic groups that are used in the sequel. They are all foundation al results.
We then deal with the Jacobian variety of a curve, the Albanese variety of an arbitrary variety, and its Picard variety, i.e., the theory of cycles of dimension 0 and codimension 1. As we shall see, the numerical theory which gives the number of points of finite order on an abelian variety, and the properties of the trace of an endomorphism are simple formal consequences of the theory of the Picard variety and of numerical equivalence. The same thing holds for the Lefschetz fixed point formula for a curve, and hence for the Riemann hypothesis for curves.
Roughly speaking, it can be said that the theory of the Albanese and Picard variety incorporates in purely algebraic terms the theory which in the classical case would be that of the first homology group. It is far from giving a complete theory of abelian varieties, and a partial list of topics which we do not discuss includes the following:
The theory of differential forms and the cohomology theory.
The infinitesimal and global theory proper to characteristic p.
The theory of linear systems and the Riemann-Roch theorem.
The theory of moduli, i.e., the classification of algebraic families of abelian varieties, and the characterization of Jacobians among abelian varieties.
Various applications to arbitrary varieties, such as, for instance, the equivalence criteria and the theorem of Neron-Severi.
Arithmetic applications like class field theory (which actually belongs to the general theory of algebraic groups) or the theorem of Mordell-Weil.
To a large extent, these topics have not reached the same state of maturity as those with which we deal in this book. Many deserve to have a whole book devoted to them. In any case, we have included at least all the results of Weil's treatise (and, of course, considerably many more).
We shall now make some remarks concerning the formal structure of the book. We begin by a list of prerequisites necessary for a rigorous understanding of the proofs given here. It should be understood, however, that much less is actually required for a general appreciation of the results stated and the methods of proofs. We hope that a good acquaintance with the language of algebraic geometry would suffice.
At the end of each chapter, we append a historical and bibliographical notice, one of whose purposes is to acquaint the reader with the current literature. We have also made comments concerning some of the directions in which the present research is leading. Further historical comment s of a more general nature have been made preceding the bibliography given at the end of the book. The index includes all the terms defined here, and the table of notation includes the symbols used most frequently. Finally, we point out that the reader who wishes to get a more detailed summary account of the content s of this book can get it by reading through the brief introductions with which we begin each chapter.
S. Lang New York, Fall 1958(Continues…)
Excerpted from "Abelian Varieties"
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Table of Contents
Chapter I Algebraic Groups,
1. Groups, subgroups, and factor groups, 1,
2. Intersections and Pontrjagin products, 6,
3. The field of definition of a group variety, 13,
Chapter II General Theorems on Abelian Varieties,
1. Rational maps of varieties into abelian varieties, 20,
2. The Jacobian variety of a curve, 30,
3. The Albanese variety, 40,
Chapter III The Theorem of the Square,
1. Algebraic equivalence, 55,
2. The theorem of the cube and the theorem of the square, 67,
3. The theorem of the square for groups, 71,
4. The kernel in the theorem of the square, 76,
Chapter IV Divisor Classes on an Abelian Variety,
1. Applications of the theorem of the square to abelian varieties, 86,
2. The torsion group, 94,
3. Numerical equivalence, 101,
4. The Picard variety of an abelian variety, 114,
Chapter V Functorial Formulas,
1. The transpose of a homomorphism, 123,
2. A list of formulas and commutative diagrams, 126,
3. The involutions, 132,
Chapter VI The Picard Variety of an Arbitrary Variety,
l. Construction of the Picard variety, 147,
2. Divisorial correspondences, 153,
3. Application to the theory of curves, 155,
4. Reciprocity and correspondences, 165,
Chapter VII The l-Adic Representations,
l. The l-adic spaces, 179,
2. Dual representations, 187,
Chapter VIII Algebraic Systems of Abelian Varieties,
1. The K/k-image, 198,
2. The generic hyperplane section, 208,
3. The K/k-trace, 211,
4. The transpose of an exact sequence, 216,
5. Duality between image and trace, 222,
6. Exact sequences of varieties, 224,
APPENDIX Composition of Correspondences,
1. Inverse images, 231,
2. Divisorial correspondences, 238,
Table of Notation, 253,