\chapter{Abelian Varieties}
The purpose of this chapter is to introduce the basic theory of abelian varieties.
We prove that the group of points on an abelian variety is always commutative as a consequence of
the rigidity lemma. We sketch the construction of the dual abelian variety (a variety that will be used
a lot in the next chapters). Finally, we discuss Jacobians of curves.
\section{Abelian Varieties over Arbitrary Fields}
Abelian varieties are the main objects of study of this paper.
\begin{defn}
An \emph{abelian variety} over a field $K$ is a smooth, proper, algebraic variety $X$ over
$K$, together with multiplication and inverse morphisms
$$
m : X \times X \ra X\ \ \textrm{(multiplication)}
$$
$$
i : X \ra X\ \ \textrm{(inverse)},
$$
and an identity element $e \in X(K)$, such that the maps $m, i$ and the element $e$ define
a group structure on $X(\overline{K})$.
\end{defn}
\begin{example}
The obvious examples are elliptic curves, since they are smooth as algebraic varieties
and have a group structure (the group law is defined by the usual addition of points law on
elliptic curves).
\end{example}
It is not clear \emph{\`a priori} whether multiplication on the group variety is commutative.
For elliptic curves, commutativity is straightforward from the definition of the group law.
To prove commutativity in general, we use the following
\begin{thm}[Rigidity Theorem]
Let $f : X \times Y \ra Z$ be a morphism of varieties over $K$. Suppose that $X$ is smooth
and there exist $y_0 \in Y(K)$ and $z_0 \in Z(K)$, such that
$$
f(X\times \{y_0\}) = \{z_0\}.
$$
Then there exists a morphism $g : Y \ra Z$, such that $f = g \circ \pi$, where
$\pi : X \times Y \ra Y$ is the projection morphism.
\end{thm}
\begin{proof}
Choose a point $x_0 \in X$ and define $g(y) = f(x_0, y)$. Choose an open affine neighborhood
$U_0$ of $z_0$ in $Z$. Since $X$ is proper over
$K$, then $\pi$ is closed. Then $W = \pi(f^{-1}(Z - U_0))$ is closed in $Y$. Then $Y - W$ is
an open set of $Y$, which is nonempty, because $y_0 \in Z - W$. Indeed,
$y \in Y - W$ if and only if $f(X \times \{y\}) \subset U_0$. Therefore, whenever $y$ is a
closed point of $X$, $f$ maps the complete variety $X \times \{y\}$ to the affine
variety $U_0$, so it must be a constant map. Therefore, for any $x \in X$ and $y \in Y$,
$$
f(x, y) = f(x_0, y) = g(y) = (g \circ \pi)(x, y).
$$
This means that $f$ and $g \circ \pi$ agree on an open dense subset of $X \times Y$ and so
they coincide everywhere.
\end{proof}
Rigidity theorem allows us to express morphisms of abelian varieties as a composition of
homomorphisms and translations.
\begin{cor}
Let $X$ and $Y$ be abelian varieties and $f : X \ra Y$ be any morphism. There is a
homomorphism $g : X \ra Y$ and $a \in Y$, such that $f(x) = g(x) + a$.
\end{cor}
\begin{proof}
Let $a = f(0)$. By replacing $f$ with $f - a$, we can assume that $f : X \ra Y$ satisfies
$f(0) = 0$. We will show that $f$ is a homomorphism. Consider $\phi : X \times X \ra Y$,
defined by $\phi(x', x'') = f(x' + x'') - f(x') - f(x'')$. For fixed $x'' \in X$,
$\phi(x',x'')$ is independent of the choice of $x'$, so $\phi(x',x'') = \phi(0, x') = 0$.
Thus, $\phi \equiv 0$ and so $f$ is a homomorphism.\footnote{Although we used additive
notation for the group law, we do not make any use of the commutativity so far.}
\end{proof}
Finally, we conclude that any abelian varieties are commutative.
\begin{cor}
If $X$ is an abelian variety, then $X$ is commutative.
\end{cor}
\begin{proof}
Consider the morphism $x \mapsto x^{-1}$. It maps the identity element to itself, so by the
previous corollary, it must be a homomorphism. Thus, $x^{-1}y^{-1} = y^{-1}x^{-1}$, so $X$
is commutative.
\end{proof}
\section{The Dual Abelian Variety in Characteristic Zero}
One of the main problems from the theory of abelian varieties
deals with studying the isomorphism classes of invertible sheaves
on the varieties (the structure of the Picard group). The goal of
this section is to endow the group of isomorphism classes of
invertible sheaves of degree 0 on $A$, considered over the closure
of $K$ (or $\textrm{Pic}^0(A_{\overline{K}})$) with the structure
of an abelian variety over $K$. We will call this variety
$A^{\vee}$ the dual of $A$ (or the Picard variety $A$).
\begin{defn}
The dual (or Picard) variety is an abelian variety $A^\vee$, together with
an invertible sheaf $\mathcal P$ on $A \times A^\vee$ (called the Poincar\'e sheaf),
such that \\
$(i)$ $\Pcal|_{\{0\} \times A^\vee}$ is trivial and for each $a \in A^\vee(\overline{K})$,
$\Pcal|_{A \times \{a\}}$ represents the element $a$. \\
$(ii)$ For every $K$-scheme $T$ and invertible sheaf $\Lcal$ on $A \times T$, such that
$\Lcal|_{\{0\} \times T}$ is trivial and $\Lcal_{A \times \{t\}}$ lies in $\textrm{Pic}^0(A_{K(t)})$
for all $t \in T(\overline{K})$, there is a unique morphism $f : T \ra A^\vee$, such that
$(1 \times f)^* \Pcal \cong \Lcal$.
\end{defn}
It follows from $(i)$ and $(ii)$ that the pair $(A^\vee, \Pcal)$, if it exists, is determined
uniquely up to unique isomorphism. Moreover, if we plug in $T = \Spc \overline{K}$, then we get
$A^\vee(\overline{K}) = \textrm{Pic}^0(A_{\overline{K}})$.
Next, we will sketch the construction of the dual variety. We start with an important result about
invertible sheaves on abelian varieties which is proved in $\S 6$ of \cite{milne:abvars}.
\begin{thm}[Theorem of the Square]
Let $A$ be an abelian variety over $K$ and $\Lcal$ be an
invertible sheaf on the variety $A$. For any point $c \in A$, we
denote by $t_c : A \ra A$ the translation map $x \mapsto c + x$.
Then
$$
t_{a+b}^* \Lcal \otimes \Lcal\cong t_a^* \Lcal \otimes t_b^*
\Lcal,
$$
for arbitrary points $a, b \in A(K)$.\footnote{By $t_c^* \Lcal$ we
mean the pullback of the sheaf $\Lcal$ under the map $t_c : A \ra
A$.}
\end{thm}
The above theorem is very important, because it can be used to
construct a homomorphism $A \ra \textrm{Pic}(A)$ in the following
way: fix an invertible sheaf $\Lcal$ on $A$ and define a map
$$
\varphi_\Lcal : A \ra \textrm{Pic}(A),\ a \mapsto t_a^*\Lcal
\otimes \Lcal^{-1}.
$$
The theorem of the square implies that
$$
t_{a+b}^*\Lcal \otimes \Lcal^{-1} \cong (t_a^*\Lcal \otimes
\Lcal^{-1}) \otimes (t_b^*\Lcal \otimes \Lcal^{-1}),
$$
so $\varphi_\Lcal$ is a homomorphism.
Next, we will show that the image of $\varphi_\Lcal$ is contained
in the subgroup Pic$^0(A)$ - the subgroup of isomorphism classes
of invertible sheaves of degree 0. To check this, it suffices to
check for any $a \in A(\overline{K})$ and $b \in A(K)$,
$$
t_a^*(\varphi_\Lcal(b)) \cong \varphi_\Lcal(b).
$$
This follows, since
$$
t_a^*(\varphi_\Lcal(b)) \cong t_a^*(t_b^*\Lcal \otimes \Lcal^{-1})
\cong t_{a+b}^*\Lcal \otimes (t_a^*\Lcal)^{-1}.
$$
The theorem of the square implies that the last sheaf is
isomorphic to $t_b^*\Lcal \otimes \Lcal^{-1} = \varphi_\Lcal(b)$.
Therefore, $\varphi_\Lcal(b) \in \textrm{Pic}^0(A)$ and we are
done.
Our next goal is to view Pic$^0(A)$ as an abelian variety, which
is a quotient of $A$. So far, we did not make any extra
assumptions about the sheaf $\Lcal$. It turns out that if $\Lcal$
is chosen to be an ample invertible sheaf and if $K$ is
algebraically closed, then $\varphi_\Lcal : A \ra
\textrm{Pic}^0(A)$ is surjective. This would allow us to endow
Pic$^0(A)$ with a structure of an abelian variety. The result is
contained in the following theorem, which follows from Prop. 10.1 of
\cite{milne:abvars}.
\begin{thm}
If $\Lcal$ is an ample invertible sheaf, then
the map $\varphi_\Lcal : A_{\overline{K}}(\overline{K}) \ra
\textrm{Pic}^0(A_{\overline{K}})$ is surjective.\footnote{By $A_{\overline{K}}$ we
mean the variety $A$, considered over $\overline{K}$, i.e. $A_{\overline{K}} \cong
A \times_K \overline{K}$.}
\end{thm}
Since abelian varieties are projective (for a complete proof, see
\cite{}), then there exists an ample invertible sheaf $\Lcal$ on
$A$. We use $\Lcal$ to define invertible sheaf $\Lcal^*$ on
$A \times A$ in the following manner
$$
\Lcal^* = m^* \Lcal \otimes \pi_1^*\Lcal \otimes \pi_2^*\Lcal^{-1},
$$
where $m : A \times A \ra A$ is the multiplication map and $\pi_1$ and $\pi_2$
are the projections on the first and the second coordinates of $A \times A$, respectively.
It follows immediately that $\Lcal^*|_{\{0\}\times A} = \Lcal \otimes \Lcal^{-1}$, which is trivial.
Moreover, $\Lcal^*|_{A \times \{a\}} = t_a^*\Lcal \otimes \Lcal^{-1} = \varphi_\Lcal(a)$, which (as we already saw)
is an element of Pic$^0(A_{\overline{K}})$. Thus, each element of Pic$^0(A_{\overline{K}})$ is represented by
$\Lcal^*|_{A \times \{a\}}$ for a finitely many $a$ (at least one such $a$). Thus, if $(A^\vee, \Pcal)$ exists, then
there is a unique isogeny $\varphi : A \ra A^\vee$, such that $(1 \times \varphi)^* \Pcal = \Lcal^*$. Furthermore,
$\varphi = \varphi_\Lcal$.
If the characteristic of $K$ is zero, then we know precisely the kernel of $\varphi_\Lcal$ as a finite group
subscheme of $A$. Indeed, it is determined by its underlying set $K_\Lcal$ with its reduced subscheme
structure. Therefore, in this case we have $A^\vee \cong A / K_\Lcal$. Moreover, $K_\Lcal$ acts on $\Lcal^*$ over
$A \times A$ by lifting the action on the second factor. If we form the quotient, we obtain a sheaf $\Pcal$, such that
$(1 \times \varphi_\Lcal)^*\Pcal = \Lcal^*$. This is pretty much the construction of $(A^\vee, \Pcal)$. A proof that this
pair satisfies the conditions in the definitions is presented in \cite{mumford:abvars}.
\section{The Dual Isogeny and the Dual Exact Sequence}
In the previous section, we explained how to dualize abelian
varieties. The next important construction is the dualization of
homomorphisms of abelian varieties.
Suppose that $f : A \ra B$ is a homomorphism of abelian varieties
and consider the induced map $f^{\vee} : \textrm{Pic } B \ra
\textrm{Pic }A$ on isomorphism classes of invertible sheaves on
$A$. Since sheaves of zero degree are mapped to sheaves of zero
degree, then we get a natural map on points $f^{\vee} : A^{\vee}
\ra B^{\vee}$, which is in fact a morphism. To give an argument
for the last statement, let $\mathcal P_B$ be the Poincar\'e sheaf
on $B \times B^{\vee}$ and consider the pullback sheaf $(f \times
1)^* \mathcal P_B$, which is a sheaf on $A \times B^{\vee}$. The
fact that $f^{\vee}$ is a morphism follows from the universal
mapping property, because $(f \times 1)^* \mathcal P_B|_{X \times
\{\tilde{y}\}}$ represents $f^{\vee}(\tilde{y})$ for any
$\tilde{y} \in Y^{\vee}$. Thus, every homomorphism of abelian
varieties induces a homomorphism on the dual varieties.
The next proposition provides a description of the dual homomorphisms
to isogenies.
\begin{prop}
Let $f : A \ra B$ be an isogeny with finite kernel $N$. Let
$N^{\vee}$ be the Cartier dual of $N$. Then the kernel of the dual
isogeny $f^{\vee} : B^{\vee} \ra A^{\vee}$ is $N^{\vee}$, i.e.
there is a short exact sequence
$$
0 \ra N^{\vee} \ra B^{\vee} \xrightarrow{f^{\vee}} A^{\vee}.
$$
\end{prop}
The proposition is proved in \cite[\S 10]{milne:abvars}. This is everything that we will need
from the theory of dual isogenies for the purpose of this project.
\section{Jacobians of Curves Over $\C$. The Analytic Construction.}
We will motivate the notion of Jacobians by looking at how they were discovered
historically. The theory of Jacobian varieties arose from the work
of Abel and Jacobi, who were studying integrals of the form
$$
I(P) = \int_{P_0}^P \omega,
$$
where $P_0$ and $P$ are points on a plane curve $C : g(x, y) = 0$ and $\omega$ is a
rational differential on $C$. The main result was the following theorem:
\begin{thm}
There is an integer $g$, depending on $C$, such that if $P_0$ is a base point and
$P_1, P_2, \dots, P_{g+1}$ are arbitrary points on $C$, then there exists
points $Q_1, Q_2, \dots, Q_g$, such that
$$
\int_{P_0}^{P_1} \omega + \dots + \int_{P_0}^{P_{g+1}} \omega=
\int_{P_0}^{Q_1} \omega + \dots + \int_{P_0}^{Q_g} \omega\ \left(\textrm{mod periods of }\int \omega\right)
$$
\end{thm}
\begin{example}
Let $C = \mathbb{P}^1$ and $\ds \omega = \frac{dx}{x}$. Then $g = 1$ and
$$
\int_{1}^{a_1} \frac{dx}{x} + \int_{1}^{b_1}\frac{dx}{x} = \int_{1}^{a_1b_1}\frac{dx}{x}.
$$
\end{example}
The theorem implies that for all $P_1, P_2, \dots, P_g, Q_1, Q_2, \dots, Q_g$, there exist
$R_1, \dots, R_g$, such that
$$
\sum_{i = 1}^g \int_{P_0}^{P_i} \omega + \sum_{i = 1}^g \int_{P_0}^{Q_i} \omega = \sum_{i = 1}^g \int_{P_0}^{R_i} \omega
$$
We recognize a group law in the last equation. The motivation behind the Jacobians is that they will be the objects that
will contain the information of how to add two such $g$-tuples $(P_1, \dots, P_g)$ and $(Q_1, \dots, Q_g)$. To realize this
in practice, we will construct a commutative algebraic group $J$, whose points will correspond to the sums
$\ds \sum_{i = 1}^g \int_{P_0}^{P_i} \omega$ and whose group law will describe precisely how we add two such sums.
To describe precisely the above idea, let $\omega$ be a rational differential on $C$ with no poles. Then
Abel's theorem (theorem 2.4.1.) can be reduced to the existence of a translation-invariant differential $\eta$ on
$J$ and a morphism of varieties $\phi : C \ra J$, such that $\phi^*\eta = \omega$. In other words,
$$
\int_{\phi(P_0)}^{\phi(P)} \eta \equiv \int_{P}^{P_0} \omega\ \left(\textrm{mod periods}\right).
$$
If one integrates all holomorphic differentials at once, we will obtain the most important of all $J$'s - the Jacobian
of the curve $J(C)$.
Although the above discussion was pretty informal, it is helpful to at least understand the idea behind the analytic
construction of the Jacobian. Since we require that $J$ contains information about the addition law for arbitrary
holomorphic differential $\omega$, then the map
$$
\phi : C \ra J(C)
$$
should set up a bijection:
$$
\phi^* : \left\{\textrm{translation invariant 1-forms on } J(C) \right\} \ra \left\{\textrm{holomorphic differentials }\omega\textrm{ on }C\right\}
$$
From here, we can conclude that dim $J(C) = \textrm{dim}_\C H^0(C, \Omega^1) = g$, where $g$ is the genus of $C$.
In order to construct $J(C)$ analytically, we have to write it as $J(C) = V / L$, where $V$ is a $\C$-vector space and
$L$ is a lattice. Let $V$ be the dual space to the space of holomorphic differentials, i.e. $V := H^0(C, \Omega^1)^*$.
The lattice $L$ will be the period lattice, i.e.
$$
L := \left\{ l \in V\ :\ l(\omega) = \int_\gamma \omega \textrm{ for some 1-cycle }\gamma\right\}
$$
In other words, $L$ can be identified with the integral homology $H_1(C, \Z)$. The map $\phi : C \ra J(C)$ will be
defined as follows: fix a base point $P_0$ and let $\phi(P)$ be the image in $V/L$ of any $l \in V$, defined by
$\ds l(\omega) = \int_{P_0}^P \omega$, where we fix a path from $P_0$ to $P$.
Since $J(C)$ is a group, it is not hard to verify that
$$
V^* \cong \left\{\textrm{transl. invariant 1-forms on }J(C)\right\} \cong H^0(C, \Omega^1),
$$
which is precisely what we want. Thus,
$$
J(C) := H^0(C(\C), \Omega^1)^* / H_1(C(\C), \Z).
$$
For more formal discussion, the reader is suggested to look at Chapter III of \cite{mumford:jacs}, or $\S 2$ of \cite{milne:jacs}.
\section{Jacobians of Curves Over Arbitrary Fields. Weil's Construction.}
This chapter only sketches Weil's construction of the Jacobian of a curve, since a
thorough discussion of then construction would be beyond the volume of this senior thesis. More details are presented
in \cite{milne:jacs}.
The formal definition of the Jacobian of a curve is an abelian variety which represents the Picard functor for that curve.
More precisely, let $C$ be a complete, nonsingular curve, defined over $k$ with positive genus
$g > 0$. One can consider the group of degree 0 divisor classes of $C$ (under linear equivalence), which we denote by
Div$^0(C)$. According to \cite[II.6]{hartshorne}, each invertible sheaf $\Lcal$ on $C$ is
of the form $\Lcal(D)$ for some divisor $D$,
and $D$ is uniquely determined up to linear equivalence. The degree of the divisor $\ds D = \sum_{i = 1}^n n_iP_i$
is defined as deg$\ds (D) = \sum_{i=1}^n n_i[K(P_i) : K]$. Hence, we can define the degree of the invertible sheaf
$\Lcal$ as deg$(\Lcal) = \textrm{deg}(D)$, where $D$ is a divisor, such that $\Lcal = \Lcal(D)$.
Let $T$ be any connected scheme over the ground field $K$ and $\Mcal$ be an invertible
sheaf on $T$. If $q : C \times T \ra T$ is the projection map on the second coordinate,
then $q^*\Mcal$ is a trivial sheaf, in the sense that $(q^*\Mcal)_t = \shf_{C_t}$ for any
$t \in T$. Therefore, we can consider the group of all invertible sheaves $\Lcal$ of
degree 0 on $C \times T$ modulo the trivial sheaves. Consider the functor
$$
P_C^0(T) := \{\Lcal \in Pic(C \times T) | deg(\Lcal_t) = 0\ \forall t \in T\} / q^*Pic(T).
$$
\begin{defn}
The Jacobian variety $J$ is an abelian variety defined over $K$, which represents the functor $P^0_C$,
whenever $C(T) \ne \varnothing$.
\end{defn}
Weil's original idea for constructing the Jacobian of a curve $C$ was to consider
the $g$-th symmetric power
$$
S^gC = C \times \dots \times C / S_g
$$
and to construct by the Riemann-Roch theorem, a partial group law on $S^gC$, i.e.
$$
m : U_1 \times U_2 \ra U_3,
$$
where $U_i \subset S^gC$ is a Zariski-open set. Then he showed that such a partial group
law extends automatically into an algebraic group $J$ with $S^gC \supset U_4 \subset J$ for some
Zariski-open $U_4$.
The formal details of Weil's construction are presented in \cite{milne:jacs}. One of the main
properties of Jacobians that we will be using quite often is that they are self-dual, i.e.
$J^\vee = J$.