Tata Institute of Fundamental Research 2012; 263 pp; hardcover ISBN10: 8185931860 ISBN13: 9788185931869 List Price: US$30 Member Price: US$24 Order Code: TIFR/13
 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 selfcontained proof of the existence of the dual abelian variety is given. The structure of the ring of endomorphisms of an abelian variety is discussed. These are appendices on Tate's theorem on endomorphisms of abelian varieties over finite fields (by C. P. Ramanujam) and on the MordellWeil 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." A publication of the Tata Institute of Fundamental Research. Distributed worldwide except in India, Bangladesh, Bhutan, Maldavis, Nepal, Pakistan, and Sri Lanka. Readership Graduate students and research mathematicians interested in abelian varieties. Table of Contents  Analytic theory
 Alegebraic 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: MordellWeil theorem by Yuri Manin
 Bibliography
 Index
