This article is in Cornell-Silverman.

This is a survey article that describes the main arithmetic properties of abelian varieties.

Here is a 17MB scan of the article in PDF format.

There are three articles on abelian varieties; the first by Rosen is
on the analytic theory, while the other two by Milne are on the geometric
theory in arbitrary characteristics and on Jacobian varieties, respectively.
These are the three main approaches to the theory of abelian varieties,
and to have all three represented in one place is very pleasant. For example,
at the elementary level, the reader can compare the proof that a connected
compact complex Lie group is commutative, in Rosen's article, with the
proof that a complete group variety is commutative, in the article "Abelian
varieties" by Milne. (Note that Section 16 describes Zarkhin's trick,
which is used in Faltings' paper.) The article "Jacobian varieties
" by Milne is a modern treatment of the subject, and as such helps
fill an important gap in the expository literature, since \n D. Mumford\en
never wrote the second volume of his book Abelian varieties \ref[Oxford
Univ. Press, London, 1970; MR 44 #219], and the book Abelian varieties
by \n S. Lang\en \ref[Interscience, New York, 1959; MR 21 #4959] was written
in the language of Weil. The bibliographical notes to Milne's article
are well worth reading; it is a pity that this kind of thing is not done
more often. Reviewed by H. Gillet |