Abelian varieties. (English) Zbl 0604.14028

Arithmetic geometry, Pap. Conf., Storrs/Conn. 1984, 103-150 (1986).
[For the entire collection see Zbl 0596.00007.]
Survey of the theory of abelian varieties over an arbitrary field, from the point of view of schemes and cohomology of coherent sheaves. A large part of the material presented here summarizes D. Mumford’s book ”Abelian varieties” (Oxford 1970; Zbl 0223.14022; 2nd edition 1974). Here are the main themes discussed: general facts about abelian varieties, the seesaw principle, the theorem of the cube and of the square, projectivity of abelian varieties, isogenies, the dual abelian variety, endomorphisms of abelian varieties, polarizations, Rosati’s involution, etc. Besides, some facts which are especially interesting for arithmetic questions are also included, e.g. the étale cohomology of abelian varieties, the zeta function of an abelian variety defined over a finite field, a weak form of the Mordell-Weil theorem, etc. This survey is very useful for those readers wanting to introduce themselves to this theory.
Reviewer: L.Bădescu


14K05 Algebraic theory of abelian varieties
14K30 Picard schemes, higher Jacobians