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\title{Weierstrass functions $\wp_{3},\wp_{5},\ldots$}
\author{V.A.Malyshev}
\date{}

\begin{document}
\maketitle

\begin{abstract}
�{\small\it The hyperelliptic curve
$$
y^{2}=(2n)!^{2}x^{2n+1}-g_{2}x^{2n-1}-g_{3}x^{2n-2}-\cdots-g_{2n+1}
$$
$\;\;\;\;\;\;$has the uniformization
$
x=\wp_{2n+1}(u),\;\; y=\wp_{2n+1}^{(2n-1)}(u).
$
}
\end{abstract}
\bigskip





\begin{center}
\large{�Contents}
\end{center}
\bigskip

\noindent {Introduction}

\medskip\noindent  1. The function $\wp_{2n+1}$

\medskip\noindent  2. Examples

\medskip\noindent {Bibliography}

%%%%%%%%%%%%%%%%%%%%%%
\section*{Introduction}

The elliptic curve
$$
y^{2}=4x^{3}-g_{2}x-g_{3}
$$
has the uniformization
$$
\begin{array}{lll}
 x&=&\wp(u),\\
 y&=&\wp'(u),\\
\end{array}
$$
where
$$
�\wp(u)=u^{-2}+c_{2}u^{2}+c_{4}u^{4}+c_{6}u^{6}+\cdots\;
$$
is the Weierstrass function~\cite{Hur1}.
\begin{center}
 \begin{picture}(400,20)
 \put(-40,30){\line(1,0){70}}
 \put(-25,15){{\footnotesize{\it 2000 Mathematics Subject Classification.}
                                      Primary 33-02, 33E05, 33E20.}}
 \put(-25,0){{\footnotesize{\it Key words and phrases.} Weierstrass functions.}}
\end{picture}
\end{center}
\newpage
\noindent Recall that
$$
\begin{array}{lll}
 {c_{2}} &=&  \frac {1}{20} \,{g_{2}}\medskip\\
 {c_{4}} &=&  \frac
 {1}{28} \,{g_{3}}\medskip\\
% {c_{6}} &=& \frac {1}{1200} \,{g_{2}}^{2}\medskip\\
 {c_{6}} &=& \frac {1}{1200} \,{g_{2}}\,{g_{2}}\medskip\\
 {c_{8}} &=&  \frac {3}{6160} \,{g_{2}}\,{g_{3}}\medskip\\
% {c_{10}} &=&  \frac {1}{156000} \,{g_{2}}^{3} +  \frac
% {1}{10192} \,{g_{3}}^{2}\medskip\\
 {c_{10}} &=&   \;\frac
 {1}{10192} \,{g_{2}}\,{g_{2}}\,{g_{2}}+\frac {1}{156000}\, {g_{3}}\,{g_{3}} \medskip\\
% {c_{12}} &=&  \frac {1}{184800} \,{g_{2}}^{2}\,{g_{ 3}}
 {c_{12}} &=&  \frac {1}{184800} \,{g_{2}}\,{g_{2}}\,{g_{ 3}}
 \medskip\\
% {c_{14}} &=& \frac {1}{21216000} \,{g_{2}}^{4} + \frac
% {3}{1905904} \,{g_{3}}^{2}\,{g_{2}}\medskip\\
 &\cdots&
\end{array}
$$
In the paper we consider the analogous uniformization for the hyperelliptic curve.




%%%%%%%%%%%%%%%%%%%%%%
\section*{1. The function $\wp_{2n+1}$}

The hyperelliptic curve
$$
y^{2}=(2n)!^{2}x^{2n+1}-g_{2}x^{2n-1}-g_{3}x^{2n-2}-\cdots-g_{2n+1}
$$
has the uniformization
$$
\begin{array}{lll}
 x&=&\wp_{2n+1}(u),\\
 y&=&\wp_{2n+1}^{(2n-1)}(u),\\
\end{array}
$$
where
$$
�\wp_{2n+1}(u)=u^{-2}+c_{2}u^{2}+c_{4}u^{4}+c_{6}u^{6}+\cdots\;.
$$
The curve
$$
y^{2}=576x^{5}-g_{2}x^{3}-g_{3}x^{2}-g_{4}x-g_{5}
$$
has the uniformization
$$
\begin{array}{lll}
 x&=&\wp_{5}(u),\\
 y&=&\wp_{5}^{(3)}(u),\\
\end{array}
$$
where
$$
\begin{array}{lll}
 {c_{2}} &=& \frac {1}{2880} \,{g_{2}}\medskip\\
 {c_{4}} &=& \frac {1}{4032} \,{g_{3}}\medskip\\
% {c_{6}} &=& \frac {1}{8640} \,{g_{4}} +  \frac
% {1}{24883200} \,{g_{2}}^{2}\medskip\\
  {c_{6}} &=&   \frac
  {1}{24883200} \,{g_{2}}\,{g_{2}}+\;\frac {1}{8640} \,{g_{4}} \medskip\\
 {c_{8}} &=&  \frac {1}{42577920} \,{g_{2}}\,{g_{3}}+ \frac {1}{19008}
 \,{g_{5}} \medskip\\
% {c_{10}} &=&
%  \frac {1}{465813504000} \,{g_{2}}^{3}
% +  \frac {1}{161740800} \,{g_{4}}\,{g_{2}} +
%  \frac {1}{211341312} \,{g_{3}}^{2}\medskip\\
  {c_{10}} &=&
   \frac {1}{465813504000} \,{g_{2}}\,{g_{2}}\,{g_{2}}
 +  \;\frac {1}{161740800} \,{g_{2}}\,{g_{4}} +
  \;\frac {1}{211341312} \,{g_{3}}\,{g_{3}}\medskip\\
 {c_{12}} &=&
 \frac {1}{551809843200} \,{g_{2}}\,{g_{2}}\,{g_{3}}  -  \frac {1}{1259089920} \,{g_{2}}\,{g_{5}}
 +  \frac {19}{4006195200}
 \,{g_{3}}\,{g _{4}}\medskip\\
 &\cdots&
\end{array}
$$
The curve
$$
y^{2}=518400x^{7}-g_{2}x^{5}-g_{3}x^{4}-g_{4}x^{3}-g_{5}x^{2}-g_{6}x-g_{7}
$$
has the uniformization
$$
\begin{array}{lll}
 x&=&\wp_{7}(u),\\
 y&=&\wp_{7}^{(5)}(u),\\
\end{array}
$$
where
$$
\begin{array}{lll}
 {c_{2}} &=&  \frac {1}{3628800} \,{g_{2}}\medskip\\
 {c_{4}} &=& \frac{1}{3628800} \,{g_{3}}\medskip\\
 {c_{6}} &=&  \;\frac {1}{8465264640000} \,{g_{2}}\,{g_{2}}+\;\frac {1}{4665600} \,{g_{4}}
   \medskip\\
 {c_{8}} &=&  \frac {1}{16094453760000} \,{g_{2}}\,{ g_{3}}+\frac {1}{13305600} \,{g_{5}}\medskip
 \\
 {c_{10}} &=&  \frac {31}{ 5590812923265024000000} \,{g_{2}}\,{g_{2}}\,{g_{2}} +
  \;\;\frac {1 }{77033908224000} \,{g_{2}}\,{g_{4}} +  \;\,\frac {1}{
 171186462720000} \,{g_{3}}\,{g_{3}}
  +  \;\frac {1}{47174400} \,{g_{6}}\medskip\\
 {c_{12}} &=&  \;\frac {1}{261636001950597120000}
 \,{g_{2}}\,{g_{2}}\,{g_{3}}+\frac {19}{5607767531520000 } \,{g_{2}}\,{g_{5}}
   +\frac {13}{4588173434880000}  \,{g_{3}}\,{g_{4}} +
    \frac {1}{140486400} \,{ g_{7}}\medskip\\
  &\cdots&
 \end{array}
$$

%%%%%%%%%%%%%%%%%%%%%%
\section*{2. Examples}


In the case
$$
y^{2}=576x^{5}-1
$$
the function $\wp_{5}$ has the form
$$
\begin{array}{l}
\wp_{5}(u) = u^{-2}  + \frac {1}{19008} u^{8} + \frac {31}{24869763072 } u^{18} + \frac
{527291}{2330058845953327104}  u^{28}\,+\,\cdots
% + \frac
%{12338895238405}{154491447173248063882433396736} u^{38}\,+\,\cdots
\end{array}.
$$
In the case
$$
y^{2}=576x^{5}-x
$$
the function $\wp_{5}$ has the form
$$
\begin{array}{l}
�\wp_{5}(u)=u^{-2}  + \frac {1}{8640} \,u^{6} +  \frac {1}{465315840} \,u^{14} + \frac
{79}{292102018560000} \,u^{22} +\,\cdots
% \frac {947191373}{
%12293181405711197798400000} \,u^{30} \,+\, {23185291171}{747563113098984791558062080000}
%\,u^{38}
\end{array}.
$$




\begin{thebibliography}{9}

\bibitem{Hur1}
A.Hurwitz, R.Courant. Theory of functions. "Nauka", Moscow, 1968.






\end{thebibliography}



\end{document}
