= Solution
The given <Weierstrass elliptic function> is even, by replacing the lattice index $\omega$ with $-\omega$. Differentiation gives
$$
\wp'(z)=-2\sum_{\omega\in\Lambda}(z-\omega)^{-3}.
$$
For any $\lambda\in\Lambda$, reindexing the sum shows $\wp'(z+\lambda)=\wp'(z)$. Thus $\wp(z+\lambda)-\wp(z)=c_\lambda$ is constant off the lattice. Replacing $z$ by $-z-\lambda$ and using evenness changes this difference to its negative, so $c_\lambda=-c_\lambda$ and $c_\lambda=0$. Hence \b[$\wp$ is doubly periodic].
It therefore descends to a <meromorphic function> on $\mathbb C/\Lambda$, with a double pole at zero modulo the lattice and no others. A <meromorphic function> is a <holomorphic map> to the <Riemann sphere>, sending poles to infinity. Locally at zero, $1/\wp(z)=z^2+O(z^4)$ is holomorphic and vanishes there, so the sphere chart at infinity supplies the required extension. The map is nonconstant because it has a pole.
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