1. Introduction

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

Given {A}, we are interested in questions of the following form: What is the structure of the {n}-torsion points {A[n]}? To compute {|A[n]|}, we are reduced to computing the degree of multiplication by {n},

\displaystyle n_A: A \rightarrow A

(which is a morphism of varieties). In fact, we will show that {n_A} is a finite flat morphism, and determine the degree of {n_A}, which is thus the cardinality of the fiber over {0}. The determination will be done by analyzing how {n_A^*} acts on line bundles. For a symmetric line bundle {\mathcal{L}} over {A}, one can prove the crucial formula

\displaystyle n_A^* \mathcal{L} \simeq \mathcal{L}^{n^2},

and comparing the Hilbert polynomials of {\mathcal{L}} and {n_A^* \mathcal{L}}, one can get as a result

\displaystyle \deg n_A = n^{2g} .

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

\displaystyle \chi(\mathcal{L}) = \frac{c_1(\mathcal{L})^g}{g!}

for any line bundle {\mathcal{L} \in \mathrm{Pic}(A)}. Consequently, in view of the asserted formula {n_A^* \mathcal{L} \simeq \mathcal{L}^{n^2}}, we find for any line bundle {\mathcal{L}}:

\displaystyle \chi(n_A^* \mathcal{L}) = n^{2g} \chi(\mathcal{L}).

Choosing {\mathcal{L}} to be a high power of a very ample line bundle, we will have {\chi(\mathcal{L}) \neq 0}. Now we can appeal to the following result:

Theorem 1 Let {X} be a proper scheme over a field. Let {G} be a finite group scheme, and let {\pi: P \rightarrow X} be a {G}-torsor. Then if {\mathcal{F}} is a coherent sheaf on {X}, we have

\displaystyle \chi(\pi^* \mathcal{F}) = (\deg \pi) \chi(\mathcal{F}).

It follows from this result that {\deg n_A = n^{2g}}. For {n} prime to the characteristic, the morphism {n_A} can be seen to be separable, and it follows as a result there are {n^{2g}} points of {n}-torsion on {A}.

2. The theorem of the cube

As stated above, an important technique in the theory of abelian varieties is the knowledge of how {n_A} acts on the Picard group of {A}. More generally, given a variety {X}, and morphisms {f, g: X \rightarrow A}, one might seek to determine {(f+g)^* : \mathrm{Pic}(A) \rightarrow \mathrm{Pic}(X)} in terms of {f^*, g^*}. There is a fairly deep theorem that allows us to do this:

Theorem 2 (Theorem of the cube) Let {f, g, h: X \rightarrow A} be three morphisms from a variety {X} to the abelian variety {A}. Then for {\mathcal{L} \in \mathrm{Pic}(A)},

\displaystyle (f + g+ h)^* \mathcal{L} \simeq (f+g)^* \mathcal{L} \otimes (f + h)^* \mathcal{L} \otimes (g + h)^* \mathcal{L} \otimes f^* \mathcal{L}^{-1} \otimes g^* \mathcal{L}^{-1} \otimes h^{*} \mathcal{L}^{-1}.

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 {n_A^*} on {\mathrm{Pic}(A)}. If we take {f = n_A, g = 1_A, h = -1_A}, we find from the theorem of the cube

\displaystyle n^*_A \mathcal{L} \simeq (n+1)_A^* \mathcal{L} \otimes (n-1)_A^* \mathcal{L} \otimes n_A^* \mathcal{L}^{-1} \otimes \mathcal{L}^{-1} \otimes (-1)_A^*\mathcal{L}^{-1}.

If we think of the function {f: \mathbb{Z} \rightarrow \mathrm{Pic}(A)} given by {f(n) = n_A^* \mathcal{L}}, we find then:

\displaystyle f(n) = f(n+1) + f(n-1) - f(n) - \mathcal{L} - (-1)_A^* \mathcal{L},

where we write the group operation additively. This allows us to work out {f} inductively, since we know {f(0), f(1), f(-1)} as {\mathcal{L}, \mathcal{O}_A, (-1)_A^* \mathcal{L}}. By induction, we find:

Proposition 3 For any line bundle {\mathcal{L}} on {A}, we have

\displaystyle n_A^* \mathcal{L} \simeq \mathcal{L}^{(n^2 + n)/2} \otimes (-1)^*\mathcal{L}^{(n^2 - n)/2} .

In particular, for a symmetric line bundle {\mathcal{L}}, we have

\displaystyle n_A^* \mathcal{L} \simeq \mathcal{L}^{n^2}.

Corollary 4 {n_A} is a finite surjective morphism, and the degree of {n_A: A \rightarrow A} is {n^{2g}}.

Proof:

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 {\mathcal{L}} on {A} (e.g., choose {\mathcal{L}_1} ample, and consider {\mathcal{L}_1 \otimes (-1)^* \mathcal{L}_1}). Then

\displaystyle n_A^* \mathcal{L} \simeq \mathcal{L}^{n^2}

is ample, and is in particular ample when restricted to the subscheme {A[n] = \ker n_A}. (This is a subscheme as the fiber over zero.) However, {n_A^* \mathcal{L}} is trivial on {A[n]}, which means that the trivial bundle on {A[n]} is ample. This means that {A[n]} is affine, hence finite.

Consequently, the map {n_A : A \rightarrow A} has finite fibers. It is also proper, so by Zariski’s Main Theorem, {n_A} is a finite morphism, necessarily surjective by dimension considerations. (If {n} was prime to the characteristic, we could see this because {n_A} would be étale.)

Finally, we need to compute the degree. Here we observe that {n_A: A \rightarrow A} 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. \Box

It’s probably worth mentioning why exactly {n_A: A \rightarrow A} makes {A} a torsor over itself (for the group scheme {A[n]}). There is an action of the group scheme {A[n]} on {A} over {A}. To say that {A \rightarrow A} is an {A[n]}-torsor is to say that we have faithful flatness (already checked) and, after pulling back, we have a splitting:

\displaystyle A \times_{n_A, A, n_A} A \simeq A \times_k A[n].

Here the fiber product means the fiber product of the two morphisms {n_A: A \rightarrow A} and {n_A: A \rightarrow A}. 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 {n} is prime to the characteristic, then {A[n] \simeq (\mathbb{Z}/n\mathbb{Z})^{2g}}.

Proof: In fact, since {n_A} acts by isomorphisms on the tangent space, {n_A} is an étale morphism. Consequently, the order of {A[n]} is the degree {n^{2g}} (in general, {A[n]} will not be reduced if {n} is not prime to the characteristic). Since this is true for any {m \mid n}, we find that we have an abelian group {G = A[n]} with the following properties:

  1. {G} has order {n^{2g} } and is {n}-torsion.
  2. Whenever {m \mid n}, the {m}-torsion in {G } has order {m^{2g}}.

By the structure theory of finite abelian groups, this gives the structure of {G = A[n]}. \Box

3. The statement about torsors

To be complete, I’d like to include a proof of the result on torsors.

Theorem 6 Let {X} be a proper scheme over a field {k}. Let {G} be a finite group scheme, and let {\pi: P \rightarrow X} be a {G}-torsor. Then if {\mathcal{F}} is a coherent sheaf on {X}, we have

\displaystyle \chi(\pi^* \mathcal{F}) = (\deg \pi) \chi(\mathcal{F}). \ \ \ \ \ (1)

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 {X} is replaced by a proper closed subscheme.

The crucial equation (1) is additive in {\mathcal{F}}, since {\pi} is flat. Consequently, if there is a short exact sequence

\displaystyle 0 \rightarrow \mathcal{F}' \rightarrow \mathcal{F} \rightarrow \mathcal{F}'' \rightarrow 0

and the result is true for {\mathcal{F}', \mathcal{F}''}, it is also true for {\mathcal{F}}. Thus, by taking filtrations of the form {\mathcal{F} \supset \mathcal{N}\mathcal{F} \supset \dots} by nilpotent ideals {\mathcal{N}}, we may assume {X} is reduced. Similarly, we may assume {X} is irreducible.

In the end, one reduces to checking it for a single sheaf {\mathcal{G}} on {X} 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 {\mathcal{G}} at the generic point. Since we know the theorem for smaller subschemes, we deduce (1) in general by (1) for {\mathcal{G}}.

In fact, we take {\pi_* \mathcal{F}} where {\mathcal{F}} is some sheaf on {P} (e.g., {\mathcal{O}_P}). The observation is that we have a cartesian square

since {\pi} is a torsor. Now we can use flat base change to observe:

\displaystyle \pi^* \pi_* \mathcal{F} \simeq \mathcal{F} \otimes \mathcal{O}_G.

In particular,

\displaystyle \chi(\pi^* \pi_*\mathcal{F} ) = (\deg \pi)\chi( \mathcal{F}) = \chi(\pi_* \mathcal{F}),

where the last equality uses the fact that {\pi} is finite and {H^\bullet(P, \mathcal{F}) \simeq H^\bullet(X, \pi_* \mathcal{F})}. \Box

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