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 1 Let 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 .
2. The theorem of the cube
As stated above, an important technique in the theory of abelian varieties is the knowledge of how acts on the Picard group of . More generally, given a variety , and morphisms , one might seek to determine in terms of . There is a fairly deep theorem that allows us to do this:
Theorem 2 (Theorem of the cube) Let be three morphisms from a variety to the abelian variety . Then for ,
This is something that deserves a (later) blog post in itself; it’s a fairly non-intuitive result, and I’m not sure at this point I have much to say. For now, I’d like to describe some of the deductions one can make.
Let’s try to inductively determine the action of on . If we take , we find from the theorem of the cube
If we think of the function given by , we find then:
where we write the group operation additively. This allows us to work out inductively, since we know as . By induction, we find:
Proposition 3 For any line bundle on , we have
In particular, for a symmetric line bundle , we have
Corollary 4 is a finite surjective morphism, and the degree of is .
In fact, if one believes the existence of ample bundles on abelian varieties (i.e., that abelian varieties are projective), then we can directly prove it. Let’s assume this, at least for now. Consider a symmetric ample line bundle on (e.g., choose ample, and consider ). Then
is ample, and is in particular ample when restricted to the subscheme . (This is a subscheme as the fiber over zero.) However, is trivial on , which means that the trivial bundle on is ample. This means that is affine, hence finite.
Consequently, the map has finite fibers. It is also proper, so by Zariski’s Main Theorem, is a finite morphism, necessarily surjective by dimension considerations. (If was prime to the characteristic, we could see this because would be étale.)
Finally, we need to compute the degree. Here we observe that has now been shown to be a finite surjective morphism; it is flat by generic flatness. One can appeal to the general theory of torsors mentioned previously, together with the Riemann-Roch formula, to get the result.
It’s probably worth mentioning why exactly makes a torsor over itself (for the group scheme ). There is an action of the group scheme on over . To say that is an -torsor is to say that we have faithful flatness (already checked) and, after pulling back, we have a splitting:
Here the fiber product means the fiber product of the two morphisms and . This can be checked on the level of functors of points, though, in which case it is just a statement about abelian groups.
Corollary 5 If is prime to the characteristic, then .
Proof: In fact, since acts by isomorphisms on the tangent space, is an étale morphism. Consequently, the order of is the degree (in general, will not be reduced if is not prime to the characteristic). Since this is true for any , we find that we have an abelian group with the following properties:
- has order and is -torsion.
- Whenever , the -torsion in has order .
By the structure theory of finite abelian groups, this gives the structure of .
3. The statement about torsors
To be complete, I’d like to include a proof of the result on torsors.
This is a fairly intuitive result: pulling back by a torsor (i.e., the analog of a finite covering space in topology) should correspondingly multiply the Euler characteristic. The actual proof is not too intuitive: it is, however, a relatively common technique. The proof is by dévissage.
Proof: By noetherian induction, we can assume that the result is true whenever is replaced by a proper closed subscheme.
The crucial equation (1) is additive in , since is flat. Consequently, if there is a short exact sequence
and the result is true for , it is also true for . Thus, by taking filtrations of the form by nilpotent ideals , we may assume is reduced. Similarly, we may assume is irreducible.
In the end, one reduces to checking it for a single sheaf on with nonzero support near the generic point. Any other sheaf can be “approximated” (up to taking a direct sum of copies) by a sum of copies of at the generic point. Since we know the theorem for smaller subschemes, we deduce (1) in general by (1) for .
In fact, we take where is some sheaf on (e.g., ). The observation is that we have a cartesian square
since is a torsor. Now we can use flat base change to observe:
where the last equality uses the fact that is finite and .