= Solution
For the <period lattice> $\Lambda=\langle\lambda,\mu\rangle$, define the <Weierstrass elliptic function> by
$$
\wp(z)=\frac1{z^2}
+\sum_{\omega\in\Lambda\setminus\{0\}}
\left(\frac1{(z-\omega)^2}-\frac1{\omega^2}\right).
$$
To prove <Normal convergence of the Weierstrass elliptic-function series>, let $K\subset\mathbb C\setminus\Lambda$ be compact and choose $R$ with $|z|\leq R$ on $K$. For $|\omega|>2R$,
$$
\left|\frac1{(z-\omega)^2}-\frac1{\omega^2}\right|
=\left|\frac{2\omega z-z^2}{\omega^2(z-\omega)^2}\right|
\leq\frac{C_K}{|\omega|^3}.
$$
The comparison series $\sum_{\omega\ne0}|\omega|^{-3}$ converges by the stated lattice-sum criterion. The <Weierstrass M-test> gives uniform convergence on $K$; since $K$ was arbitrary, the series converges locally uniformly away from $\Lambda$. The <locally uniform convergence of holomorphic functions> makes its sum holomorphic there, and the displayed principal term gives a double pole at every lattice point.
The function $\wp$ is even and satisfies
$$
\wp(z)=\wp(w)
\quad\Longleftrightarrow\quad
z\equiv\pm w\pmod\Lambda.
$$
The three nonzero <two-torsion point of a complex torus>[two-torsion points] $z_1,z_2,z_3$ are distinct modulo $\Lambda$ and each equals its own negative modulo $\Lambda$. Therefore no two can have the same $\wp$-value, so $e_1,e_2,e_3$ are distinct. Equivalently, $\wp$ is a degree-two branched map and each half-lattice point already occupies its fiber with multiplicity two.
For a transitive example, take the <equianharmonic lattice>
$$
\Lambda=\mathbb Z+\mathbb Z\omega,
\qquad \omega=e^{2\pi i/3},
$$
and define $\theta([z])=[\omega z]$. Multiplication by $\omega$ preserves $\Lambda$, so $\theta:\mathbb C/\Lambda\to\mathbb C/\Lambda$ is a <conformal equivalence>. On the nonzero half-lattice points it acts by
$$
\left[\frac12\right]
\longmapsto\left[\frac\omega2\right]
\longmapsto\left[\frac{1+\omega}{2}\right]
\longmapsto\left[\frac12\right],
$$
because $\omega^2=-1-\omega$. Thus it acts transitively.
Back to article page
