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Abelian Varieties
David Mumford, Brown University, Providence, RI
A publication of the Tata Institute of Fundamental Research.
Tata Institute of Fundamental Research
2012; 263 pp; hardcover
ISBN-10: 81-85931-86-0
ISBN-13: 978-81-85931-86-9
List Price: US$30
Member Price: US$24
Order Code: TIFR/13
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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 scheme-theoretic 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. These 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."

A publication of the Tata Institute of Fundamental Research. Distributed worldwide except in India, Bangladesh, Bhutan, Maldavis, Nepal, Pakistan, and Sri Lanka.


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: Mordell-Weil theorem by Yuri Manin
  • Bibliography
  • Index
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