Past exam of the geometry course of the University of Oxford / 2011 / 2

For a nonconstant holomorphic map $f:X\to Y$ between compact connected Riemann surfaces, let $n$ be its degree and $e_p$ its local degree at $p$. The Riemann-Hurwitz formula isEquivalently, $2g_X-2=n(2g_Y-2)+{\sum }_p(e_p-1)$. Only finitely many terms are nonzero; their sum is the total ramification. In local coordinates the map has the form $w=z^{e_p}$ after suitable coordinate changes.

Let $\Lambda =\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2$ and $T=\mathbb{C}/\Lambda$. Its fundamental parallelogram with opposite sides identified is a torus, so $\chi (T)=0$. The Weierstrass elliptic function is even and doubly periodic. One way to verify the latter from its series is to differentiate: the locally normally convergent derivative series iswhich is absolutely convergent away from the lattice and periodic by reindexing. Thus $\wp (z+{\omega }_j)-\wp (z)$ is constant. Evaluating at $z=-{\omega }_j/2$ and using evenness shows that this constant is zero for each generating period.

Therefore $[z]\mapsto \wp (z)$ descends to a meromorphic function on $T$, with one double pole at $[0]$. Viewed as a map to the Riemann sphere, a pole is a holomorphic point in the target coordinate $1/w$. The map is consequently holomorphic everywhere and

The Riemann-Hurwitz formula gives total ramification four because $0=2\cdot 2-R$. The double pole at $[0]$ has local degree two and contributes one. Since ${\wp }^{\prime }$ is odd and periodic, it vanishes at each nonzero point satisfying $[h]=[-h]$. These areThey are three distinct finite points, each contributing at least one to ramification. Together with $[0]$ they exhaust the total of four. Each therefore has local degree exactly two, meaning its derivative has a simple zero. There are no other zeros of ${\wp }^{\prime }$, and their translates by $\Lambda$ give the full list in the complex plane.

Multiplication by two on the complex torus is a holomorphic unramified coveringIts kernel is $(\frac{1}{2}\Lambda )/\Lambda$, of order four; its derivative is the nonzero constant two. Therefore $\mathrm{deg}[2]=4$. Composition with the degree-two Weierstrass elliptic function givesIt is well-defined because changing $z$ by $\omega \in \Lambda$ changes $2z$ by the period $2\omega$.

A pole occurs exactly when $2z\in \Lambda$. Thus the poles on $T$ areEach is a double pole, since the multiplication-by-two map is locally unramified and $\wp$ has a double pole at $[0]$. There are four distinct poles, each of order two, consistent with the degree eight.

Put $R(z)=\frac{1}{4}({\wp }^{\prime \prime }/{\wp }^{\prime }{)}^2-2\wp (z)$. This is an elliptic function. Its only possible poles modulo $\Lambda$ are $0$ and the three simple zeros $h$ of ${\wp }^{\prime }$. At such a nonzero half-period, periodicity and evenness give $\wp (h+t)=\wp (h-t)$, soIt follows that $R(h+t)=1/(4t^2)+O(1)$ with no simple-pole term. This is exactly the principal part of $\wp (2h+2t)=\wp (2t)$.

At zero the source expansion yieldsConsequently $R(z)=1/(4z^2)+O(z^2)$, again matching $\wp (2z)$. Thus $R(z)-\wp (2z)$ is a holomorphic function on a compact connected Riemann surface, hence constant. Its limit at zero is zero, provingThe equality extends over the apparent singularities as an identity of meromorphic functions.