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\begin{document}

\section*{Problem Set 8}
Let $\omega_1,\omega_2$ be two complex numbers, linearly independent over $\RR$, and $\Lambda=\ZZ\langle \omega_1,\omega_2\rangle \subset \CC$ be the lattice they generate.
\begin{enumerate}[leftmargin=0.7cm]
\item Show that the series $$\tfrac{1}{z^2}+\textstyle \sum_{\omega\in \Lambda}' \left(\tfrac{1}{(z-\omega)^2}-\tfrac{1}{\omega^2} \right)$$ defining $\wp(z)$ is absolutely and uniformly convergent on any compact subset of $\CC$ which does not contain any of the points of $\Lambda$. Try to do this without the Lemma at the end of Lecture 22. (Perform a direct estimate.)
\item There are three perspectives on the addition theorem for $\wp$: analytic, geometric, and algebraic.  I'll do the latter two in class and you'll provide the analytic approach here, by following the steps below. Consult Ahlfors pp.~276-77 for hints.

\noindent (a) $\wp(z)-\wp(u)=-\tfrac{\sigma(z-u)\sigma(z+u)}{\sigma(z)^2\sigma(u)^2}$

\noindent (b) $\tfrac{\wp'(z)}{\wp(z)-\wp(u)}=\zeta(z-u)+\zeta(z+u)-2\zeta(z)$

\noindent (c) $\zeta(z+u)=\zeta(z)+\zeta(u)+\tfrac{1}{2}\tfrac{\wp'(z)-\wp'(u)}{\wp(z)-\wp(u)}$

\noindent (d) $\wp(z+u)=-\wp(z)-\wp(u)+\tfrac{1}{4}\left(\tfrac{\wp'(z)-\wp'(u)}{\wp(z)-\wp(u)}\right)^2$ [addition formula]

\noindent (e) $\wp(2z)=\tfrac{1}{4}\left(\tfrac{\wp''(z)}{\wp'(z)}\right)^2-2\wp(z)$
\item For this problem, assume that $\Lambda$ is such that $g_2=-4$ and $g_3=0$. (That is, $\CC/\Lambda$ is isomorphic to the algebraic curve $y^2=4x^3+4x$.  Such a $\Lambda$ exists simply by taking it to be the set of all periods of $\tfrac{dx}{y}$ on this curve.)

\noindent (a) Express the RHS of formula 2(e) as a rational function of $\wp(z)$.

\noindent (b) Let $u$ be such that $\wp(u)=1$  (and $\wp'(u)=2\sqrt{2}$). Show that $u$ is 4-torsion, i.e.~$4u\in \Lambda$.

\noindent (c) Let $u$ be such that $\wp(u)=\tfrac{p}{2^a q}$, where the fraction is written in lowest terms, $a$ is an odd natural number, and $p,q$ are odd integers. Show that $u$ is of ``infinite order'', i.e.~no integer multiple of it lies in $\Lambda$. [Hint: put $u_0:=u$. Show that $\wp(u_1)$, where $u_1=2u_0$, is of the same form, with bigger $a$, and iterate. Then suppose $u_0$ was $N$-torsion and produce a contradiction via the pigeonhole principle.]
\item This exercise concerns the Theta function, defined in Lecture 22.  (See also Lecture 23 for $\eta_1$.)

\noindent (a) Check that $\theta(z+1)=\theta(z)$.

\noindent (b) Check that $\theta(z)=C e^{h(z)}\sigma(z+\tfrac{\tau+1}{2})$, where $h(z)=-\tfrac{\eta_1}{2}z^2-(\tfrac{\eta_1\tau}{2}+\tfrac{\eta_1}{2}+\pi i)z$.
\end{enumerate}

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