This, like the previous and probably the next few posts, is an attempt at understanding some of the ideas in Mumford’s *Abelian Varieties.*

Let be an abelian variety. Last time, I described a formula which allowed us to express the pull-back of a line bundle as

This was a special case of the so-called “theorem of the cube,” which allowed us to express, for three morphisms , the pull-back

in terms of the various pull-backs of partial sums. Namely, we had the formula

In this formula, we take and constant maps at , respectively. Let , etc. denote the translation maps on . Then are the relevant sums, and we have

We get the theorem of the square:

Theorem 1 (Theorem of the square)Let be an abelian variety, , and points. Then

In the rest of this post, I’ll describe some applications of this result, for instance the fact that abelian varieties are projective. (more…)