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This is problem 2.1 in O. Forster, Lectures on Riemann Surfaces, Springer-Verlag, 1981. Let $\Gamma\in\mathbb{C}$ be a lattice. Let $p_\Gamma(z)=\frac{1}{z^{2}}+\sum_{\omega\in\Gamma\backslash\{0\}}(\frac{1}{(z-\omega)^{2}}-\frac{1}{\omega^{2}}) $. Let $f$ be a meromorphic function that is doubly periodic function with respect to $\Gamma$ which has its poles at points of $\Gamma$ and which has he following Laurant expansion about the origin $f(z)=\sum_{k=-2}^\infty c_kz^k$ where $c_{-2}=1,c_{-1}=c_0=0$ Prove that $f=p_\Gamma$.

All I know so far is that $p_\Gamma$ satisfies $p_\Gamma'^2=4p_\Gamma^3-g_2p_\Gamma-g_3$.

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To show $f = \wp_\Gamma$, show that $\wp_\Gamma - f \equiv 0$.

You know that $\wp_\Gamma - f$

  • is doubly periodic with respect to $\Gamma$, because $\wp_\Gamma$ and $f$ are.
  • has no poles except possibly in the lattice points, because $\wp_\Gamma$ and $f$ have poles only in the lattice points.
  • actually has no poles at all, since the principal part of $f$ in $0$ equals the principal part of $\wp_\Gamma$ in $0$.

Deduce that $\wp_\Gamma - f$ is constant. Use the further information about the Laurent expansion of $f$, and that of $\wp_\Gamma$ (which you need to obtain from the given partial fraction decomposition of $\wp_\Gamma$) to deduce that that constant is $0$.

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