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%% Yoichi Imayoshi
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\title[Holomorphic sections of Riemann surfaces]
{A remark on holomorphic sections of certain holomorphic families
of Riemann surfaces}
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%% Authors
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\author[Y.~Imayoshi]{Yoichi Imayoshi}
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\address[Yoichi Imayoshi] {Department of Mathematics, Osaka City University,
Su\-gi\-mo\-to, Su\-mi\-yo\-shi-ku, Osaka 558-8585, Japan}
\email{imayoshi@sci.osaka-cu.ac.jp}
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\author[N.~Nogi]{Toshihiro Nogi}
\address[Toshihiro Nogi] {Department of Mathematics, Osaka City
University, Su\-gi\-mo\-to, Su\-mi\-yo\-shi-ku, Osaka 558-8585,
Japan}
\email{nogicchi@ezweb.ne.jp}
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%\date{\today}
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%% Abstract
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\begin{abstract} In this paper,
we study a special holomorphic family $(M, \pi, R)$ of closed
Riemann surfaces of genus two over a fourth punctured torus $R$, which is a
kind of a Kodaira surface and is constructed by Riera. We give two explicit
defining equations for $(M, \pi, R)$ by using elliptic functions, and determine all
the holomorphic sections of $(M, \pi, R)$.
Proofs will appear elsewhere.
\end{abstract}
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%% AMS Mathematics Subject Classification
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%
\subjclass{Primary 11G30, Secondary 14H15, 32G15, 32J25}
\keywords{holomprhoc section,
holomorphic family of Riemann surfaces,
function field, Kodaira surface, moduli space, Teichm\"uller space,
Diophantine equation}
\maketitle
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%% Section 1 : Introduction
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{Introduction}
%
Let us consider a Diophantine equation
\begin{equation}
\sum_{i+j+k=N} A_{ijk}X^i Y^j Z^k = 0
\end{equation}
over the function field $K$ of a closed Riemann surgace $\hat{R}$.
Here $K$ is the field of all meromorphic functions on $\hat{R}$ and
the coefficients $A_{ijk}$ are elements of $K$. The problem is to
find the solutions $(X, Y, K) \in P^2(K)$ of the function equation
(1) over $\hat{R}$. \par
This problem is reformulated geometrically in the following way.
It is assumed that we find a Zariski open subset $R$ of $\hat{R}$ and
a Zariski open subset $M$ of the algebraic surface $\hat{M}$ defined by
$$
\hat{M} = \{([x,y,z], t) \in P^2(\mathbf{C}) \times \hat{R} \, | \,
\sum_{i+j+k=N} A_{ijk}(t) x^i y^j z^k = 0 \}
$$
such that the holomorphic map $\pi \colon M \to R$ given by
$\pi([x,y,z], t) = t$ satisfies the two conditions:
\begin{itemize}
\item[(1)] $\pi$ is of maximal rank at every point of $M$, and
\item[(2)] for every $t \in R$, the fiber $S_t = \pi^{-1}(t)$ of $M$ over $t$ is
a Riemann surface of fixed finite type $(g,n)$, where $g$ is the genus of $S_t$ and
$n$ is the number of punctures of $S_t$.
\end{itemize}
We call such a triplet $(M, \pi, R)$ is a {\em holomorphic family of Riemann surfaces}
of type $(g, n)$ over $R$.\par
We assume throughout this paper that a holomorphic family $(M, \pi, R)$ of
Riemann surfaces is of type $(g, n)$ with $2g - 2 + n > 0$ and
the base surface $R$ is of finte type, i.e., a Riemann surface obtained by removing
at most a finite number of points from a closed Riemann surface. \par
Two holomorphic families $(M_1, \pi_1, R)$ and $(M_2, \pi_2, R)$ of Riemann
surfaces are called {\em isomorphic} if there exists a biholomorphic map
$f\colon M_1 \to M_2$ with $\pi_1 = \pi_2 \circ f$. A holomorphic family
$(M, \pi, R)$ of Riemann surfaces is {\em locally trivial} if for every point
$p_0 \in R$ there exsits a neighborhood $U_0$ of $p_0$ in $R$ such that
$(\pi^{-1} (U_0), \pi |\pi^{-1} (U_0), U_0)$ is isomorphic to the trivial
holomorphic family $(U_0\times S_{p_0},\pi_0, U_0)$, where
$\pi_0 \colon U_0\times S_{p_0} \to U_0$ is the canonical projection.
It is known that a holomorphic family $(M, \pi, R)$ of Riemann surfaces is
locally trivial if and only if the fibers $S_t$ are all isomorphic.\par
A holomorphic map $s \colon R \to M$ is said to be a {\em holomorphic section}
of a holomorphic family $(M, \pi, R)$ of Riemann surfaces if s stisfies
$\pi \circ s = \operatorname{id}$ on $R$. Note that if $(X, Y, K) \in P^2(K)$
satisfies the function equation (1), then $s(t) = ([X(t), Y[t], Z[t] ,t)$ gives rise to a
holomorphic section of $(M, \pi, R)$. \par
By using theory of Teichm\"uller space, Imayoshi and Shiga in [6] gave a proof for a finiteness
theorem of sections (Mordell conjecture) and a finiteness theorem of families (Shafarevich conjecture). See also Arakelov [1], Faltings [3], Grauert [4], Jost and Yau [7],
Manin [9], McMullen [10], and Parshin [11]. \par
In this paper we deal with a special holomorphic family $(M, \pi, R)$ of closed
Riemann surfaces of genus two over a fourth punctured torus $R$, which is a
kind of a Kodaira surface as [8] and is constructed by Riera in [12]. We give two explicit
defining equations for $(M, \pi, R)$ by using elliptic functions, and determine all
the holomorphic sections of $(M, \pi, R)$. \par
%
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%% Section 2 : Construction of a holomorphic family
%% of Riemann Riemann surfaces
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{Construction of a certain holomorphic family $(M, \pi, R)$ of Riemann surfaces}
%
Now we explain briefly a construction of our
holomorphic family $(M, \pi, R)$ of Riemann surfaces, which is due to
Riera [12]. \par
Take a point $\tau$ in the upper half-plane $\mathbf{H}$ in the complex plane $\mathbf{C}$. Let $\Gamma_{1, \tau}$ be the discrete subgroup of
$\operatorname{Aut}(\mathbf{C})$ generated by two translations $z \mapsto z+1$ and
$z \mapsto z+\tau$. Denote by $\hat{T}$ a torus defiend by the
quotient space $\mathbf{C}/\Gamma_{1, \tau} = \{[z] \, | \, z \in \mathbf{C}\}$.
We set $p_0 = [0] \in \hat{T}$ and $T = \hat{T} \setminus \{p_0 \}$.\par
For a point $p \in T$ we take two replicas of the torus $\hat{T}$ cut along
a simple arc from $p$ to $p_0$, and call them sheet I and sheet II.
The cut on each sheet has two edges, labeled $+$ edge and $-$ edge. To construct a Riemann surface $X_p$, we attach the $+$ edge
on sheet I and the $-$ edge on sheet II, and then attach the $+$ edge on sheet II and
the $-$ edge on sheet I. Then we obtain a closed Riemann surface $X_p$ of genus
two and the two-sheeted covering $X_p \to \hat{T}$ which is branched
over $p_0$ and $p$ with branch order $2$. It should be noted that
the above procedure depends, of course, not only on the choice of the point $p$
but also on the choice of the \lq\lq cut\rq\rq\,from $p$ to $p_0$. Essentially we can take
four different \lq\lq cuts\rq\rq\, $\alpha_1, \alpha_2, \alpha_3$, and $\alpha_4$
between $p$ and $p_0$ (see Fig.1). \par
To specify the \lq\lq cut\rq\rq\, \,we construct a four-sheeted unbranched covering
\begin{equation}
\rho \colon R \to T
\end{equation}
of $T$ such that $R$ is a torus with four punctures as follows:
Let $\Gamma_{2, 2\tau}$ be the discrete subgroup of
$\operatorname{Aut}(\mathbf{C})$ generated by two translations $z \mapsto z+2$ and
$z \mapsto z+2 \tau$. Denote by $\hat{R}$ a torus defiend by the
quotient space $\mathbf{C}/\Gamma_{2, 2\tau} = \{[z] \, | \, z \in \mathbf{C}\}$.
Let $\hat{\rho} \colon\hat{R} \to \hat{T}$ be the canonical projection
given by $\hat{\rho} ([z]) = [z]$. We set $R = \hat{\rho}^{-1}(T)$ and
$\rho = \hat{\rho}| R$. The good thing is that a point $t = [z] \in R$ determines a point
$p = \rho ([z]) \in T$ and a \lq\lq cut\rq\rq\, $\alpha = \hat{\rho}(\beta)$
from $p$ to $p_0 = [0]$, where $\beta$ is a simple arc on $\hat{R}$ from $[0]$ to $t$.
Denote by $S_t$ the closed Riemann surface of genus two which is
a two-sheeted branched covering surface of $\hat{T}$ constructed by a
\lq\lq cut\rq\rq\, $\alpha = \hat{\rho}(\beta)$. Note that the
two-sheeted branched covering $\Pi_t \colon S_t \to \hat{T}$ is uniquely determined by the choice of $t \in R$ and does not depend on $\beta$. \par
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%@Fig.1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\centering
\includegraphics[scale=1,clip]{Fig1.eps}
\caption{
For each $p \in T$, there are precisely four two-sheeted branched
coverings $S_{t_i} \to \hat{T}$, where $t_i \in R$ with $\hat{\rho}(t_i) = p$ and
the cut $\alpha_i$ is given by the projection $\hat{\rho}(\beta_i)$ of a simple arc
$\beta_i$ between $[0]$ and $t_i$ on $\hat{R}$.}
\label{Fig.1}
\end{figure}
Now we have the following theorem (see Riera [12]):
\begin{theorem} In the above situation, let
\begin{align*}
M &= \bigsqcup_{t\in R} \{t\} \times S_t,\\
\pi &\colon M \to R,\, \, \pi (t,q) =t.
\end{align*}
Then $(M, \pi, R)$ is a holomorphic family of closed Riemann surfaces
of genus two over a fourth punctured torus $R$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Section 3: Defining equations
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{Defining equations for $(M, \pi, R)$}
In this section we find defining equations for $(M, \pi, R)$.
For any point $t = [\,\tilde{t}\,] \in R$,
Abel's theorem shows there exists a meromorphic function $f_{t}$ on $\hat{T}$
which has two zeros $[0]$ and $\rho(t)$ of order one, and a pole $q_t = \rho(t)/2$
of order two. Moreover in order to determine $f_{t}$ uniquely,
we assume that $(df_{t}/dz)([0])=1$. This function $f_t$ is given explicitly
as follows (see Clemens [2]):
\begin{align}
f_{t}([z]) &= \frac{1}{\theta'\left (1/2 + \tau/2 \right )}
\times \frac{\theta\left(-\tilde{t}/2+1/2+
\tau/2 \right )^2}
{\theta\left (-\tilde{t} +1/2+\tau/2\right )}\times \notag\\
& \hskip 4.8mm \frac{\theta \left (z +1/2+\tau/2 \right ) \,
\theta \left (z - \tilde{t} + 1/2+\tau/2\right )}
{\theta \left (z- \tilde{t}/2 +1/2 + \tau/2\right )^2}.
\end{align}
Here the theta function $\theta (z, \tau)$ is defined by
$$
\theta (z, \tau) = \sum_{k=-\infty}^{\infty} e^{\pi i (k^2 \tau + 2kz)},
\quad z \in \mathbf{C}.
$$
Then we have the following assertion:
\begin{theorem} In the above situation, let
\begin{align*}
M_E &=\{(t, p, w)\in R\times\hat{T}\times\hat{\bf C}~|~ w^{2}=f_t (p) \},\\
\pi_E&: M_E \to R,\,\, \pi_E(t, p, w) = t.
\end{align*}
Then the triplet $(M_E, \pi_E, R)$ is a holomorphic family of closed Riemann surfaces of genus two, and it is isomorphic to $({\mathcal M}, \pi, R)$ in Theorem 1.
\end{theorem}
We find another defining equation for $(M, \pi, R)$.
The holomorophic map $f_t \colon \hat{T} \to \hat{\mathbf{C}}$ has
four branch points $q_t$ (pole), $a(t), b(t)$, and $c(t)$, where
\begin{align*}
a(t) &= f_t([(\tilde{t}+1)/2]),\\
b(t) &= f_t([(\tilde{t}+\tau)/2]),\\
c(t) &= f_t([(\tilde{t}+1 + \tau)/2]).
\end{align*}
Let $g_t$ be the meromorphic function on $\hat{T}$ of degree $3$ satisfying
\begin{itemize}
\item[(1)] $g_t$ has simple zeros $[(\tilde{t}+1)/2], [(\tilde{t}+\tau)/2],
[(\tilde{t}+1 + \tau)/2]$,
\item[(2)] $g_t$ has a pole $[\tilde{t}]$ of order $3$, and
\item[(3)] $g_t([0]) = i$.
\end{itemize}
This function $g_t$ is given by
\begin{align*}
g_{t}(z) = i e^{-2\pi i z}\times &
\frac{\theta\left(-\tilde{t}/2 +1/2 + \tau/2 \right )^3}
{\theta \left (-\tilde{t}/2\right ) \,
\theta \left ( -\tilde{t}/2+1/2 \right ) \,
\theta \left ( - \tilde{t}/2 +\tau/2 \right )}
\times \\
& \frac{\theta \left (z - \tilde{t}/2 \right ) \,
\theta \left (z -\tilde{t}/2 +1/2 \right ) \,
\theta \left (z - \tilde{t}/2 +\tau/2 \right )}
{\theta \left (z- \tilde{t}/2 + 1/2 + \tau/2 \right )^3}.
\end{align*}
Setting $x=f_t, y=g_t$, we have a functional relation
\begin{equation}
y^2 = \frac{1}{a(t) b(t) c(t)}\, (x - a(t))\, (x - b(t))\,(x - c(t))
\end{equation}
on $\hat{T}$.
Now we have the following theorem:
\begin{theorem} In the above situation, let
\begin{align*}
P_t(x) &= ( x^2 - a(t) )( x^2 - b(t) )( x^2 - c(t) ),\\
M_{HE} &= \{(t, x, y)\in R\times\hat{\bf C}\times\hat{\bf C} \, | \, y^{2}= P_t(x)\},\\
\pi_{HE}&\colon M_{HE} \to R,\,\, \pi_{HE}(t, x, y) = t.
\end{align*}
Then the triplet $(M_{HE}, \pi_{HE}, R)$ is a holomorphic family of
closed Riemann surfaces of genus two, and it is
isomorphic to $(M, \pi, R)$ in Theorem 1.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Section 4: Holomorphci sections $(M, \pi, R)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{Holomorphci sections $(M, \pi, R)$}
Let us study the holomorphic sections of
$(M, \pi, R)$ in Theorem 1. The following is our main theorem
in this paper.\par
\begin{theorem} The holomorphic family $(M, \pi, R)$ of
closed Riemann surfaces of genus two in Theorem 1 has
exactly two holomorphic sections $s_1, s_2$.
These sections are given by $s_1(t) = (t, p_0 ) $ and $s_2(t) = (t, \rho(t) )$
for every $t\in R.$
\end{theorem}
In order to prove this theorem, we need the following two theorems (cf. Imayoshi [5], Theorem 4 and Theorem 5):
\begin{theorem} The holomorphic family $(M, \pi, R)$ in Theorem 1 has a canonical
completion $(\hat{M}, \hat{\pi},\hat{R})$, where $\hat{M}$ is a compact two dimensional normal complex analytic space and $\hat{\pi} \colon \hat{M} \to \hat{R}$ is holomorphic.
Moreover every holomorphic section $s\colon R \to M$ has a holomorphic extension
$\hat{s} \colon \hat{R} \to \hat{M}$.
\end{theorem}
\begin{theorem} The holomorphic map $\Pi \colon M =
\bigsqcup_{t\in R} \{t\} \times S_t \to \hat{T}$ defined by
$\Pi(t, q) = \Pi_t(q)$ has a holomorphic extension
$\hat{\Pi} \colon \hat{M} \to \hat{T}$.
\end{theorem}
Theorem 6 is proved by Theorems 2, 3, and 5.\par
Now we can prove Theorem 4 as follows: Let $s \colon R \to M$ be an
arbitrary holomorphic section of $(M, \pi, R).$ Theorem 5 and 6 imply that
the holomorphic map $\varphi = \Pi \circ s \colon R \to \hat{T}$ has a
holomorphic extention $\hat{\varphi} = \hat{\Pi} \circ\hat{s} \colon
\hat{R} \to \hat{T}$. Let $\tilde{\varphi} \colon \mathbf{C} \to \mathbf{C}$ is
a lift of $\hat{\varphi} \colon \hat{R} \to \hat{T}$. Then $\tilde{\varphi}(z) =
Az + B, z \in \mathbf{C}$ for some constants $A, B \in \mathbf{C}$.
Since $\varphi = \Pi \circ s$, we can show that we may assume that
$A = 0, B=0$, or $A=1, B=0$. In the case $A = 0, B=0$, we have
the section $s_1(t) = (t, p_0)$, and in the case $A = 1, B=0$, we have
the section $s_2(t) = (t, \rho[t])$.\par
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%% References
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%% your references, and try to use labels for the bibitems, which
%% are uniquely assigned to you in order to avoid conflicts with other authors.
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\end{thebibliography}
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