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

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.