= Solution
For a degree-$d$ nonconstant <holomorphic map> from a compact connected <Riemann surface> of <genus> $g$ to the <projective line>, the <Riemann-Hurwitz formula> is
$$
\boxed{2g-2=-2d+\sum_{p\in X}(e_p-1),}
$$
where $e_p$ is the <local degree>. A <meromorphic function> defines a map to the <sphere> by giving its ordinary finite values away from poles and value infinity at each pole. The reciprocal function is holomorphic in a neighborhood of a pole, proving holomorphicity there. For use in Riemann-Hurwitz the function must be nonconstant; a nonzero constant still defines a <holomorphic map>, but of degree zero.
Let $X=\mathbb C/\Lambda$. The <double pole> of $\wp$ gives a degree-two map to the <sphere>. The <torus> has <genus> one, so total ramification is four. One simple <ramification point> is zero, above infinity. At the three nonzero half-period classes $h_i$, evenness and periodicity give $\wp'(h_i)=0$. Each is a <ramification point> of <local degree> two, and these account for all remaining ramification. Put $e_i=\wp(h_i)$. They are distinct, since a degree-two fiber cannot contain two points each of <local degree> two.
At each $h_i$, $\wp-e_i$ has a double zero and $\wp'$ has a simple zero. Consequently the elliptic quotient
$$
R(z)=\frac{\wp'(z)^2}{(\wp(z)-e_1)(\wp(z)-e_2)(\wp(z)-e_3)}
$$
has removable singularities there and no zeros or poles elsewhere except possibly zero. Near zero, $\wp=z^{-2}+O(z^2)$ and $\wp'=-2z^{-3}+O(z)$, so $R\to4$. Thus $R$ is holomorphic on the compact <torus> and constant, proving
$$
\boxed{\wp'^2=4(\wp-e_1)(\wp-e_2)(\wp-e_3).}
$$
\b[The printed additional requirement that all three $e_i$ be nonzero is false.] For the square lattice $\Lambda=\mathbb Z+i\mathbb Z$, invariance of the defining series under multiplication by $i$ gives $\wp(iz)=-\wp(z)$. At $h=(1+i)/2$, one has $ih=h-1$, so periodicity gives $\wp(h)=\wp(ih)=-\wp(h)$ and hence $\wp(h)=0$. This is one of the three branch values. Distinctness, which is all the cubic construction needs, remains valid.
Define
$$
\Phi(z)=[\wp(z):\wp'(z):1]\quad(z\notin\Lambda),\qquad\Phi(0)=[0:1:0].
$$
The image lies on $Y^2Z=4(X-e_1Z)(X-e_2Z)(X-e_3Z)$. Distinct roots make its affine part smooth, and the point at infinity is smooth because the derivative of its equation with respect to $Z$ is nonzero there. Near zero the projective coordinates $X/Y=\wp/\wp'\sim-z/2$ and $Z/Y=1/\wp'\sim-z^3/2$ extend holomorphically. Every finite $X/Z$ value has two preimages under $\wp$ except at a branch value. Evenness pairs those preimages as $z,-z$, and oddness of $\wp'$ distinguishes them unless they are the same half-period class. Therefore the two coordinates together give injectivity and reach both signs of $Y$, proving surjectivity onto the cubic. Thus \b[$\Phi$ is a bijective <holomorphic map> to a nonsingular cubic], including when one branch value is zero.
Back to article page
