\chapter{Jacobians of Curves}
\section{Jacobians of Curves Over $\C$. The Analytic Definition.}
\subsection{Motivation For Studying Jacobians.} 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
$$
\int_{P_0}^{P_1} \omega + \dots + \int_{P_0}^{P_{g+1}} \omega=
\int_{P_0}^{Q_1} \omega + \dots + \int_{P_0}^{Q_g} \omega.
$$
\end{thm}
\begin{thm}[Abel]
Let $P_1, \dots, P_r \in C(\C)$ and $Q_1, \dots, Q_r \in C(\C)$. There exists a
meromorphic function on $C(\C)$ if and only if for any paths $\gamma_i$ from $P_0$
to $P_i$ and $\gamma_i'$ from $P_0$ to $Q_i$ there exists $\gamma \in H_1(C(\C), \Z)$,
such that
$$
\sum \int_{\gamma_i} \omega - \sum \int_{\gamma_i'} \omega = \sum \int_{\gamma} \omega,\
\forall \omega \in \Gamma(C(\C), \Omega^1).
$$
\end{thm}
\begin{thm}[Jacobi]
Suppose that $l : \Gamma(C(\C), \Z) \ra \C$ is a linear function. There exists $g$ points
$P_1, \dots, P_g \in C(\C)$ and paths $\gamma_i$ from $P_0$ to $P_i$, such that
$l(\omega) = \sum \int_{\gamma_i} \omega$ for all $\omega \in \Gamma(C(\C), \Omega^1))$.
\end{thm}
\subsection{Construction of the Jacobian}
\section{Definition of Jacobians of Curves over Arbitrary Fields.}
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$^0(C)$. But, [Hartshorne],
divisor classes correspond to invertible sheaves and Div$^0(C)$ is isomorphic to
the group Pic$^0(C)$ of invertible sheaves on $C$ of degree 0.
Roughly speaking, the Jacobian of $C$ is an abelian variety, which contains the
data of Pic$^0(C)$. More precisely, suppose that $T$ is a connected scheme over $k$
and $\Lcal$ is an invertible sheaf on $C \times_k T$. One can think of $\Lcal$ as a
family of sheaves $\{\Lcal_t\}_{t \in T}$, where $\Lcal_t := \Lcal|_{C \times \{t\}}$.
The first observation is that the degree of each $\Lcal_t$ is independent of $t$.
\begin{prop}[Mum-1, \S5]
The function $t \mapsto deg(\Lcal_t)$ is locally constant.
\end{prop}
\verb Proof: For each $n$, by the Riemann-Roch theorem ***cite***, we have
$$
\chi(C_t, \Lcal^n_t) = deg(\Lcal_t^n) + 1 - g_t = n \cdot deg(\Lcal_t) + 1 - g_t,
$$
where $g_t$ is the genus of
the fiber curve $C_t$. Thus, if $\chi(C_t, \Lcal^n)$ is written as a polynomial in $n$, then
deg $\Lcal_t$ is simply the leading coefficient. The statement will follow then from the
following technical lemma:
\begin{lem}
Suppose that $X$ and $T$ are Noetherian schemes, $T = \Spc A$ is affine, and $f : X \ra T$
is a proper morphism. Suppose further that $\Lcal$ is a coherent sheaf on $X$, which is
flat over $T$. Then for every integer $k > 0$, the function
$t \mapsto \textrm{dim}_{k(y)}H^p(X_t, \Lcal_t)$ is upper-semicontinuous.
\end{lem}
\verb Proof: ***later*** ... $\hfill \Box$
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. Let
$$
P_C^0 := \{\Lcal \in Pic(C \times T) | deg(\Lcal_t) = 0\ \forall t \in T\} / q^*Pic(T).
$$
Before defining the Jacobian of $C$ abstractly, as an abelian variety, having certain
properties, we need one more definition.
\begin{defn}
Suppose that $X$ and $S$ are $k$-schemes. Define
$X(S) := \textrm{Mor}_k(S, X)$. Notice that in the case $S = \Spc k$, $X(k)$ is
precisely the set of $k$-rational points of $X$. One often says that $X$ represents
the functor $S \mapsto X(S)$, which maps $k$-schemes to sets.
\end{defn}
\begin{thm}
There exists an abelian variety $J$ and a morphism of functors
$\iota : P_C^0 \ra J$, such that whenever $C(T)$ is nonempty, then
$\iota : P_C^0(T) \ra J(T)$ is an isomorphism. Moreover, the pair $(J, \iota)$
is unique up to unique isomorphism. The variety $J$ is called the Jacobian of the
curve $C$.
\end{thm}
Note that the theorem implies the existence of the Jacobian of a curve. Since it will
take a bit of work to construct $J$, we will postpone the proof for the next
sections. The uniqueness of the Jacobian will be a consequence of the following universal
property:
\begin{thm}
The Jacobian of $C$ can be characterized with the following universal property: suppose
that $P \in C(k)$ is a fixed $k$-rational point. Then there exists a unique invertible
sheaf $\Mcal$ on $C \times J$, such that $\Mcal |_{C \times \{0\}}$ and
$\Mcal|_{\{P\} \times J}$ are both trivial and such that for any invertible sheaf
\end{thm}
The goal is to derive theorem 2 as a consequence of theorem 1.
\section{Abelian Varieties Are Quotients of Jacobian Varieties.}
\section{Jacobians of Modular Curves. Computational Examples.}