Appendix: Geometric theory

Summary

Many of the results in Chapter III can be stated in purely geometric language, even though the proofs given there are analytic. The object of this appendix is to state how far they can be regarded as geometric theorems, by outlining without proofs some of the geometric theory of abelian varieties. The language used will be that of classical algebraic geometry; for an account in scheme-theoretic language see Mumford [8].

Let V be a variety, not necessarily complete, non-singular or irreducible. V is called a group variety if there is given a group law on the points of V such that the structure maps V × V → V and V → V which respectively take u × v to uv and v to v⁻¹ are regular maps in the sense of algebraic geometry. We say that a field k is a field of definition for V considered as a group variety if V, the two structure maps and the identity element of the group are all defined over k. In this case V must be non-singular, the irreducible component of V which contains the identity element is a normal subgroup of V and thereby a group variety defined over k, and the other irreducible components of V are cosets of this subgroup; for this reason we could without much loss have required V to be irreducible, and some writers do so.