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 ${G = A[n]}$. $\Box$

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 ${H^\bullet(P, \mathcal{F}) \simeq H^\bullet(X, \pi_* \mathcal{F})}$. $\Box$