We now turn to discuss about the higher dimensional analogue of elliptic curves: abelian varieties. Most of results from elliptic curves presented in the book of Silverman can be generalized to abelian varieties, with more abstract proofs. In this post, we will prove some fundamental results in the theory of elliptic curves based on the algebro-geometric techniques.
Theorem 0. Let $E_1, E_2$ be elliptic curves defined over a field $k$, and $\alpha: E_1\to E_2$ is a regular map, then $\alpha$ is a composition of a homomorphism and a translation map.
Theorem 1. Let $E$ be an elliptic curve defined over an algebraically closed field $k$. Denote $n: E\to E$ the multiplication by $n$ map, then $\deg n=n^2$, and $n$ is an unramified map iff $(n,char(k))=1$. And in this case, $E[n]$-the $n$-torsion subgroup of $E$ has $n^2$ elements and $E[n]\cong \mathbb{Z}_n\oplus \mathbb{Z}_n$.
1. Rigidity theorem. First, let us define what an abelian variety is. One can see for elliptic curves, the map $m: E\times E\to E$ sends $(p,q)$ to $p+q$ and $i: E\to E$ send $p\to -p$ are given by polynomials, and they are actually regular maps, and it satisfies the group law on $E$ with an identity element $0\in E$. Now, an abelian variety is defined as a complete connected variety $A$ such that there exists regular maps $m: A\times A\to A$, $i: A\to A$ together with an element $0\in A$ such that $(m,i,0)$ satisfies the group law in $A$.
From the definition, an abelian variety is automatically smooth, for the set of non-singular points on $A$ is an open dense subset. If we choose a point $p\in A$ is smooth, then for all point $q\in A$, the translation map $\tau_{-p+q}$, is an automorphism of $A$. This yields $q$ is also a smooth point of $A$. Hence, $A$ is a smooth variety. An abelian variety is defined to be a complete variety, but it can be proved that $A$ is projective. This result is due to Weil.
We should recall the definition of an elliptic curve $E$ over a field $k$: it is a non-singular projective curve of genus 1 with at least one rational point, denoted $O$. The Riemann-Roch's theorem implies that there exists a bijective map $E\to Pic^0(E)$ by sending $P\to [P]-[O]$, and the group structure on $E$ is induced naturally from the group structure on $Pic^0(X)$. It is abelian. However, it is difficult to prove the converse.
Theorem 1.1. A projective curve with group law defined by regular maps is actually an elliptic curve.
Proof. The proof is presented on Milne's note on abelian varieties, introduction pages. It is an amazing proof, where one has to use Lefschetz's fixed point formula, for etale cohomology. It is a very motivated proof for those who wants to study etale cohomology (including me). (Q.E.D)
As we can see, the group law on an abelian variety of dimension 1 (elliptic curve) turns out to be commutative. This also holds for higher dimension abelian varieties.
Theorem 1.2 (Rigidity Theorem). Let $U,V,W$ be varieties, with $V$ is complete, $V\times W$ is connected and $\alpha: V\times W\to U$ is a regular map. Assume that there exists $u_0\in U, v_0\in V,w_0\in W$ such that $\alpha(\{v_0\}\times W)=\{u_0\}=\alpha(V\times \{w_0\})$, then $\alpha(V\times W)=\{u_0\}$.
Proof. (See Milne's note on Abelian Varieties-Theorem 1.1)
By using Rigidity Theorem, we will prove the general version of Theorem 0.
Corollary 1.3. Let $\alpha: A\to B$ be a regular map between abelian variety. Then $\alpha$ is a composition of a translation and a homomorphism.
Proof. By a suitable translation, we can assume that $\alpha(0)=0$. Consider the map $h: A\times A\to B$ sending $(a_1,a_2)$ to $\alpha(a_1+a_2)-\alpha(a_1)-\alpha(a_2)$. The restriction of $h$ to $\{0\}\times A$ and $A\times \{0\}$ is just $0$. This yields $h(A\times A)=\{0\}$ by rigidity theorem, and $\alpha$ is a homomorphism (Q.E.D)
Using this, we can prove that the group law on an abelian variety.
Corollary 1.4. The group law on an abelian variety $A$ is commutative.
Proof. We now use a trick: a group $A$ is abelian iff the map $-1: a\mapsto -a$ is a homomorphism. Using Theorem 1.3, -1 is a regular map (by definition), which sends $0$ to $0$, and hence, it is a homomorphism. This yields $A$ is an abelian group. (Q.E.D)
2. Theorem of the cube. We now turn to the main theorem in our notes. The theorem of the cube will be one of the main tools for studying line bundles on abelian varieties.
Theorem 2.1 (Theorem of the cube). Let $U, V, W$ be complete irreducible varieties, and $L$ is a line bundle on $U\times V\times W$. Assume that there exists $u_0\in U,v_0\in V,w_0\in W$ such that $L$ restricts on $\{u_0\}\times V\times W, U\times \{v_0\}\times W, U\times V\times \{w_0\}$ are trivial. Then $L$ is trivial.
Now, it has the following important corollary
Corollary 2.2. Let $A$ be an abelian variety and $L$ is a line bundle on $A$. Let us denote $p_{123}:A\times A\times A\to A$ the map sends $(p,q,r)$ to $p+q+r$, $p_{12}: A\times A\times A\to A$ sends $(p,q,-)$ to $p+q$ (similarly for $p_{23}, p_{13}$), and $p_1:A\times A\times A\to A$ sends $(p,-,-)$ to $p$ (similarly for $p_2,p_3$). Then
$$L':=p_{123}^*L\otimes p_{12}^*L^{-1}\otimes p_{23}^*L^{-1}\otimes p_{13}^*L^{-1}\otimes p_1^*L \otimes p_2^*L \otimes p_3^*L$$
is a trivial line bundle.
Proof. We recall that if $\pi: L\to V$ is a line bundle and $f: W\to V$ is a regular map, then the pullback bundle of $L$ is defined $f^*L:=\{(w,l)\in W\times L| f(w)=\pi(l)\}$, and $f^*L$ is a line bundle on $W$ together with $\pi': f^*L\to W$ the projection map into the first coordinate.
Now, one can restrict $L'$ to $B:=\{0\}\times A\times A$, this can be seen
$$p_{123}^*L=\{(a,b,c,l)\in A\times A\times A\times L|a+b+c=\pi(l)\}$$
And $p_{123}^*L|_B=\{(0,b,c,l)\in \{0\}\times A\times A\times L|b+c=\pi(l)\}=p_{23}^*L$. This yields $p_123^*L\otimes p_{23}^*L^{-1}$ is trivial on $B$.
Also
$$p_{12}^*L=\{(a,b,-,l)\in A\times A\times A\times L|a+b=\pi(l)\}$$
And $p_{12}^*L|_B=\{(0,b,-,l)\in A\times A\times A\times L|b=\pi(l)\}=p_2^*L$. This yields $p_{12}^*L^{-1}\otimes p_2^*L$ is trivial on $B$. The similar result holds for $p_{13}L^{-1}\otimes p_3^*L$.
Finally, it can be seen that $p_1^*L|_B=\{(0,-,-,l)\in A\times A\times A\times L|\pi(l)=0\}$ is also a trivial bundle on $B$. This shows $L'$ is a trivial bundle on $B$. Due to the symmetry of the index, $L'$ is also trivial on $A\times \{0\}\times A$ and $A\times A\times \{0\}$. By theorem of the cube, $L'$ is trivial. (Q.E.D)
Corollary 2.3. Let $f,g,h$ be regular maps from a variety $V$ to an abelian variety $A$, and $L$ is a line bundle on $A$. Then
$$L':=(f+g+h)^*L\otimes (f+g)^*L^{-1}\otimes (g+h)^*L^{-1}\otimes (h+f)^*L^{-1}\otimes f^*L\otimes g^*L\otimes h^*L$$
is a trivial bundle.
Proof. Let us define the map $V\xrightarrow{(f,g,h)} A\times A\times A$. Then $L'$ is just the pullback bundle of the line bundle on the previous corollary. And the pull back of a trivial bundle is a trivial bundle. (Q.E.D)
Using this, we can prove the important
Corollary 2.4. Let $A$ be an abelian variety and $n: A\to A$ the multiplication by $n$ map. Then
$$n^*L\cong L^{(n^2+n)/2}\otimes (-1)^*L^{(n^2-n)/2}$$
In particular, if $L$ is a symmetric line bundle, i.e. $L\cong (-1)^*L$, then $n^*L\cong L^{n^2}$, and if $L$ is anti-symmetric, i.e. $L\cong (-1)^*L^{-1}$, then $n^*L\cong L^n$.
Proof. We just use the previous corollary for $(f,g,h)=(n,1,-1)$ and use induction on $n$. (Q.E.D)
We are now ready for the proof of Theorem 1.
Theorem 1. Let $E$ be an elliptic curve defined over an algebraically closed field $k$. Denote $n: E\to E$ the multiplication by $n$ map, then $\deg n=n^2$, and $n$ is an unramified map iff $(n,char(k))=1$. And in this case, $E[n]$-the $n$-torsion subgroup of $E$ has $n^2$ elements and $E[n]\cong \mathbb{Z}_n\oplus \mathbb{Z}_n$.
Proof. We recall that if $f: X\to Y$ is a non-constant regular map between smooth projective curves, then for all $Q\in Y$, we have pull-back of divisor $f^*[Q]=\sum_{P\in Y, f(P)=Q}e_{f,P}[P]$, where $e_{f,P}$ is the ramification index at $P$ with respect to $f$. Let $D$ be a symmetric divisor on $E$, i.e. $(-1)^*D=D$ (for example, one can choose $D=[O]$). Then it can be seen by the previous corollary that $n^*[O]\sim n^2[O]$.
Also, $n^*[O]=\sum_{P\in E[n]}e_{n,P}[P]$, and $(-n)^*[O]=\sum_{P\in E[n]}e_{-n,P}[P]$. Because $-1$ is an automorphism of $E$, $e_{n,P}=e_{-n,P}$, i.e. $n^*[O]=(-n)^*[O]$. But then, $(-n)^*=(-1)^*n^*$. And this yields
$$(-1)^*(n^*[O])=(-1)^*(\sum_{P\in E[n]}e_{n,P}[P])=\sum_{P\in E[n]}e_{n,P}[-P]$$
From this, one can see $e_{n,P}=e_{n,-P}$, and hence $\sum_{P\in E[n]}e_{n,P}P=O$. And we know that $\deg(n)=\sum_{P\in E[n]}e_{n,P}$. This implies $n^*[O]-\deg(n)[O]$ is a principal divisor, i.e. $n^*[O]\sim \deg(n)[O]$. Now, we can see $n^2[O]\sum \deg(n)[O]$. This yields $\deg n=n^2$. This proves our first statement.
For the second statement, the map $\alpha: A\to B$ between abelian varieties will induce the map $d\alpha: T_{A,0}\to T_{B,0}$, where $T_{A,0}$ is the tangent space of $A$ at $0$. And it can be seen that $\alpha\mapsto d\alpha$ is a homomorphism. This yields $dn$ is actually the multiplication by $n$ map on the tangent space of $A$ at $0$. It will be $0$ iff $char(k)$ divides $n$. Otherwise, it is an isomorphism. Now, we use the fact that if $f: X\to Y$ is a finite map between varieties, and $f(x)=y$, then $f$ is unramified at $x$ iff the induced map on tangent space $T_{X,x}\to T_{Y,y}$ is injective (Stack Project, Lemma 32.16.8). This yields $n$ is unramified at 0 iff $(n,char(k))=1$. Because the translation map is an automorphism (of curves), this will induce an isomorphism on tangent spaces. Hence, $n$ is unramified iff $(n,char(k))=1$.
With the assumption that $(n,char(k))=1$, the multiplication map by $n$ is unramified, i.e. $e_{n,P}=1$ for all $P\in E[n]$. This yields $\#E[n]=n^2$. By using the fundamental theorem of finite abelian group, we will obtain $E[n]\cong \mathbb{Z}_n\oplus \mathbb{Z}_n$ (Q.E.D)
Remark. The more general version of Theorem 1 for abelian varieties can be found in Milne's note (Theorem 7.2).
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