Past exam of the geometry course of the University of Oxford / 2008 / B / 7 / 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 iswhere $e_p$ 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 ${\wp }^{\prime }(h_i)=0$. 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 ${\wp }^{\prime }$ has a simple zero. Consequently the elliptic quotienthas removable singularities there and no zeros or poles elsewhere except possibly zero. Near zero, $\wp =z^{-2}+O(z^2)$ and ${\wp }^{\prime }=-2z^{-3}+O(z)$, so $R\to 4$. Thus $R$ is holomorphic on the compact torus and constant, provingThe 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.

DefineThe image lies on $Y^{2}Z=4(X-e_{1}Z)(X-e_{2}Z)(X-e_{3}Z)$. 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 $X/Y=\wp /{\wp }^{\prime }\sim -z/2$ and $Z/Y=1/{\wp }^{\prime }\sim -z^3/2$ 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 ${\wp }^{\prime }$ 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 $\Phi$ is a bijective holomorphic map to a nonsingular cubic, including when one branch value is zero.