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Abelian Varieties

David Mumford
Hindustan Book Agency
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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When I was a graduate student, this book was never in the library. In the Science Library, it was perpetually “on call”; if you were lucky enough to get your hands on it, you had to return it soon because someone else wanted it. Most of the time, however, it was just not there, and sometimes the library didn’t know where it was.

There was a departmental library as well, but Abelian Varieties couldn’t be found there either. Rumor had it that every time a copy was put in that library, it was stolen. On top of all that, the book was out of print!

There was one sign of hope: the book could still be obtained in India. One of my colleagues, who was from India and went home one summer, agreed to buy a bunch of them. I still have that copy.

Abelian varieties are one of the natural generalizations of elliptic curves: higher-dimensional algebraic varieties that are also (abelian) algebraic groups. They are fairly classical objects, going back all the way to the work on algebraic functions by Abel and Jacobi. Here they are treated from the point of view of algebraic geometry.

The book is based on lectures given by David Mumford at the Tata Institute for Fundamental Research in 1967-68. Two appendices, by C. P. Ramanujan and by Y. Manin, were added in the second edition in 1974. A few mistakes were corrected in this newly-typeset printing, but it is essentially the same as the 1974 version.

When Mumford won the Steele Prize for Exposition, the committee listed this book as one of the reasons. They said it “remains the definitive account of the stubject.” I think that is right. 

This new edition is much prettier, with better typesetting, a nicer (hard) cover. Most importantly, it can be bought from the AMS. So perhaps now it’ll find its way to all those libraries that could never get a copy and to the personal libraries of all those aspiring number theorists and algebraic geometers who wanted one.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.


  • 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