Functorial Formulas
We first define the transpose of a homomorphism, i.e., the contravariant mapping induced on the Picard varieties. We prove that the transpose of an exact sequence (up to isogenies) is exact (up to isogenies).
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Lanu
ABELIAN VARIETIES
Springer Science+Business Media, LLC
Serge Lang Yale University Department of Mathematics Box 2155 Yale Station New Haven, Connecticut 06520 U.S.A.
AMS Subject Classification: 14KXX
Library of Congress Cataloging in Publication Data Lang, Serge, 1927Abelian varieties. Bibliography: p. lncludes index. 1. Abelian varieties. 1. Title. QA564ยท L277 19 83 5 12 '.55
Published in 1983 by Springer Science+Business Media New York Originally published by Springer-Verlag New York Inc in 1983
987654321
ISBN 978-0-387-90875-5 DOI 10.1007/978-1-4419-8534-7
ISBN 978-1-4419-8534-7 (eBook)
PREFACE TO THE SPRINGER EDITION
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. Mumford, 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 concerning 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. Therefore I thank Springer-Verlag for the opportunity to keep the book in print. S. LANG
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FOREWORD Pour des simplifications plus subs tantielles, 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. 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 co dimension 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
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