Abelian Varieties

This is a reprinting of the revised second edition (1974) of David Mumford's classic 1970 book. It gives a systematic account of the basic results about abelian varieties. It includes expositions of analytic meth­ods applicable over the ground field of complex numbers, as well as of scheme­theoretic methods used to deal with inseparable isogenies when the ground field has positive characteristic. A self ­contained proof of the existence of the dual abelian variety is given. The structure of the ring of endomorphisms of an abelian variety is discussed. There are ap­pendices on Tate's theorem on endomorphisms of abelian varieties over finite fields (by C. P. Ramanujam) and on the Mordell ­Weil theorem (by Yuri Manin).

David Mumford was awarded the 2007 AMS Steele Prize for Math­ematical Exposition.

According to the citation, "Abelian Varieties ... remains the definitive account of the subject ... the classical theory is beautifully intertwined with the modern theory, in a way which sharply illuminates both ... [It] will remain for the foreseeable future a classic to which the reader returns over and over".

Contents

Introduction
Analytic Theory
Algebraic Theory Via Varieties
Algebraic Theory Via Schemes
Hom(X,X) and l-adic Representation
Appendix I: The Theorem of Tate by C.P. Ramanujam
Appendix II: Mordell-Weil Theorem by Yuri Manin
Bibliography
Index