**1. Introduction**

Let be an abelian variety over an algebraically closed field , of dimension . One of the basic tools in analyzing the properties of is the study of line bundles on . It’s a little non-intuitive to me why this is the case, so I’m going to try to motivate the topic.

Given , we are interested in questions of the following form: What is the structure of the -torsion points ? To compute , we are reduced to computing the degree of multiplication by ,

(which is a morphism of varieties). In fact, we will show that is a finite flat morphism, and determine the degree of , which is thus the cardinality of the fiber over . The determination will be done by analyzing how acts on line bundles. For a *symmetric* line bundle over , one can prove the crucial formula

and comparing the Hilbert polynomials of and , one can get as a result

Another way of phrasing this deduction is the following. By the Hirzebruch-Riemann-Roch formula and the parallelizability of an abelian variety, we have

for any line bundle . Consequently, in view of the asserted formula , we find for any line bundle :

Choosing to be a high power of a very ample line bundle, we will have . Now we can appeal to the following result:

Theorem 1Let be a proper scheme over a field. Let be a finite group scheme, and let be a -torsor. Then if is a coherent sheaf on , we have

It follows from this result that . For prime to the characteristic, the morphism can be seen to be separable, and it follows as a result there are points of -torsion on . (more…)