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Could anyone help me out a little here? My homework question reads:

Let $\Gamma \subset \mathbb{C}$ be a lattice. The Weierstrass $\wp$-function is defined as $$ \wp_\Gamma(z) = \sum_{\omega \in \Gamma \ \backslash \{ 0 \} }\frac{1}{(z - \omega)^2} - \frac{1}{\omega^2}. $$

  1. Prove that $\wp_\Gamma$ is a well defined meromorphic function on the torus $\pi: \mathbb{C} \rightarrow \mathbb{C} / \Gamma$.
  2. Prove that if $f \in \mathcal{M}(\mathbb{C}/ \Gamma) \cap \mathcal{O}((\mathbb{C}/ \Gamma) \backslash [0])$ has a pole of order $2$ at $[0]$, has an expansion $$f \circ \pi(z) = \sum_{j = -2}^{\infty} c_j z^j, c_{-2} = 1, c_{-1} = c_0 = 0,$$ then $f = \wp_\Gamma$.

I've never seen the Weierstrass $\wp$-function defined as above. I've only seen it as: $$ \wp_\Gamma^{original}(z) = \frac{1}{z^2} + \sum_{\omega \in \Gamma \ \backslash \{ 0 \} }\frac{1}{(z - \omega)^2} - \frac{1}{\omega^2}. $$ I take it that this is because $\wp_\Gamma(z)$ takes elements from $\mathbb{C}/\Gamma$ and not from $\mathbb{C}$ as is usual. Is this correct?


To show that $\wp_\Gamma(z)$ is meromorphic, does it go along these lines?

Let $\Gamma = \mathbb{Z}\omega_1 + \mathbb{Z}\omega_2 = \{ n\omega_1 + m\omega_2 : n,m \in \mathbb{Z} \}$, where $\omega_1, \omega_2 \in \mathbb{C}$ are linearly independent over $\mathbb{R}$.

Clearly, $\wp_\Gamma(z) \in \mathcal{O}((\mathbb{C}/\Gamma) \backslash [0])$.

If $z_1 \in \mathbb{C}$, $\pi(z_1) = [z_1] \in \mathbb{C}/\Gamma$ s.t. $[z_1] \cap [0] \neq \emptyset$ then $z_1 = n_1\omega_1 + m_1\omega_2$ for $n_1,m_1 \in \mathbb{Z}$, $\omega_1,\omega_2 \in \Gamma$, and $$ \wp_\Gamma(z) = \sum_{\omega \in \Gamma \ \backslash \{ 0 \} }\frac{1}{(z - \omega)^2} - \frac{1}{\omega^2} = \frac{1}{(z - (n_1\omega_1 + m_1\omega_2))^2} - \frac{1}{(n_1\omega_1 + m_1\omega_2)^2} + \sum_{\omega \in \Gamma \ \backslash \{(n_1,m_1),0 \} }\frac{1}{(z - \omega)^2} - \frac{1}{\omega^2} $$

has a pole of order $2$ at $z_1$.

Then it follows that $\wp_\Gamma(z)$ is meromorphic on $\mathbb{C}/\Gamma$.


For well-definedness, is this approach correct?

Let $z_1, z_2 \in \mathbb{C}$ s.t. $[z_1] \cap [z_2] \neq \emptyset$. Then $z_1 = z_2 + n\omega_1 + m\omega_2$, for $n,m \in \mathbb{Z}$. Then show that the choice of representative of the equivalence class doesn't matter (i.e. $\wp_\Gamma(z_1) = \wp_\Gamma(z_2)$)?

Second part

For the second part, I don't have a clue. Could anyone help me get started?

Any help greatly appreciated!

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