Abelian Varieties

ISBN-13: 9780387908755 / Angielski / Miękka / 1983 / 256 str.

Serge Lang; S. Lang

Opis

A belian Varieties has been out of print for a while. Since it was written, the subject has made some great advances, and Mumford's book giving a scheme theoretic treatment has appeared (D. Mum ford, Abelian Varieties, Tata Lecture Notes, Oxford University Press, London, 1970). However, some topics covered in my book were not covered in Mumford's; for instance, the construction of the Picard variety, the Albanese variety, some formulas concern ing numerical questions, the reciprocity law for correspondences and its application to Kummer theory, Chow's theory for the K/k-trace and image, and others. Several people have told me they still found a number of sections of my book useful. There fore I thank Springer-Verlag for the opportunity to keep the book in print. S. LANG v FOREWORD Pour des simplifications plus subs tan tielles, Ie developpement futur de la geometrie algebrique ne saurait manquer sans do ute d' en faire apparaitre. It is with considerable pleasure that we have seen in recent years the simplifications 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 foundational results.

Spis treści

I Algebraic Groups.- 1. Groups, subgroups, and factor groups.- 2. Intersections and Pontrjagin products.- 3. The field of definition of a group variety.- II General Theorems on Abelian Varieties.- 1. Rational maps of varieties into abelian varieties.- 2. The Jacobian variety of a curve.- 3. The Albanese variety.- III The Theorem of the Square.- 1. Algebraic equivalence.- 2. The theorem of the cube and the theorem of the square.- 3. The theorem of the square for groups.- 4. The kernel in the theorem of the square.- IV Divisor Classes on an Abelian Variety.- 1. Applications of the theorem of the square to abelian varieties.- 2. The torsion group.- 3. Numerical equivalence.- 4. The Picard variety of an abelian variety.- V Functorial Formulas.- 1. The transpose of a homomorphism.- 2. A list of formulas and commutative diagrams.- 3. The involutions.- VI The Picard Variety of an Arbitrary Variety.- 1. Construction of the Picard variety.- 2. Divisorial correspondences.- 3. Application to the theory of curves.- 4. Reciprocity and correspondences.- VII The l-Adic Representations.- 1. The l-adic spaces.- 2. Dual representations.- VIII Algebraic Systems of Abelian Varieties.- 1. The K/k-image.- 2. The generic hyperplane section.- 3. The K/k-trace.- 4. The transpose of an exact sequence.- 5. Duality between image and trace.- 6. Exact sequences of varieties.- Appendix Composition of Correspondences.- 1. Inverse images.- 2. Divisorial correspondences.- Table of Notation.