= Solution
A covering map is locally a disjoint union of homeomorphisms onto the base. A space is simply connected when it is path-connected and every loop is null-homotopic. By the <uniformization theorem>, the simply connected Riemann surfaces are
$$
\mathbb C_\infty,\qquad\mathbb C,\qquad\mathbb D
$$
up to analytic isomorphism.
A lattice is a discrete <subgroup>
$$
L=\mathbb Z\omega_1\oplus\mathbb Z\omega_2\subset\mathbb C
$$
with $\omega_1/\omega_2\notin\mathbb R$. The <Weierstrass elliptic function>
$$
\wp_L(z)=\frac1{z^2}+
\sum_{0\ne\omega\in L}
\left(\frac1{(z-\omega)^2}-\frac1{\omega^2}\right)
$$
converges normally on compact subsets of $\mathbb C\setminus L$, defines a meromorphic map $\mathbb C\to\mathbb C_\infty$, and is nonconstant because it has a double pole at every lattice point. Reindexing the normally convergent <derivative> <series> shows that $\wp_L'$ is $L$-periodic; evenness fixes the integration constants, so $\wp_L(z+\omega)=\wp_L(z)$.
The invariance makes
$$
\overline\wp([z])=\wp_L(z)
$$
well-defined and unique on $\mathbb C/L$. It is analytic because the quotient projection is locally biholomorphic.
Neither map is a covering. For any nonzero $\omega\in L$, periodicity and oddness of $\wp_L'$ give
$$
\wp_L'(\omega/2)=\wp_L'(-\omega/2)=-\wp_L'(\omega/2),
$$
so $\wp_L'(\omega/2)=0$. The map is not a local homeomorphism there; since the quotient projection is locally biholomorphic, the descended map fails for the same reason.
There is no covering $\mathbb C/L\to\mathbb C_\infty$. The torus is connected, while every connected covering of the simply connected sphere is a homeomorphism. A torus is not homeomorphic to a sphere, for example because their fundamental <groups> are $\mathbb Z^2$ and $0$.
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