Past exam of the geometry course of the University of Oxford / 2009 / A / 2

For a nonconstant holomorphic map of degree $d$ between compact connected Riemann surfaces, the Riemann-Hurwitz formula isHere $e_p$ is the local degree, characterized by local coordinates in which the map is $z\mapsto z^{e_p}$. The sum is finite and its terms are nonnegative; a ramification point has $e_p>1$.

If $g_X=0$, the Riemann-Hurwitz formula gives $2=d(2-2g_Y)-R$, where $R\ge 0$. If $g_Y\ge 1$, the right side is nonpositive, a contradiction. Therefore

The given Weierstrass elliptic function is even, by replacing the lattice index $\omega$ with $-\omega$. Differentiation givesFor any $\lambda \in \Lambda$, reindexing the sum shows ${\wp }^{\prime }(z+\lambda )={\wp }^{\prime }(z)$. 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 $\wp$ 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.

The total pole order in one period cell is two, so the map has degree two. The Riemann-Hurwitz formula for genus one to genus zero gives $0=2\cdot 2-R$, hence $R=4$. Each local degree is at most two, so every ramification contribution is exactly one. ThereforeThey are zero and the three nonzero half-period classes. Zero is ramified because its reciprocal has a double zero. At a nonzero half-period, oddness and periodicity of ${\wp }^{\prime }$ force ${\wp }^{\prime }=0$; the count shows these exhaust the remaining points.

For a degree-two map to the Riemann sphere, the Riemann-Hurwitz formula gives $2-2g=4-R$. Again each ramification index is two, so the number of ramification points equals $R$. Thus