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".
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
2008, 9788185931869, 263 pages, hard cover, Rs: 400.00