Past exam of the algebraic curves course of the University of Oxford / 2023 / 3 / d

An explicit answer, initially written where $yz\ne 0$, is the meromorphic differential on a Riemann surfaceThe ratios $x/z$ and $y/z$ are functions on $C$, so the expression does not depend on the representative of its homogeneous coordinates. It is a scalar multiple of an invariant differential on an elliptic curve.

On the complex torus $\mathbb{C}/\Lambda$, the differential $dw$ descends from $\mathbb{C}$ because translation by any element of the period lattice leaves $dw$ unchanged. It is holomorphic and nowhere zero. The given biholomorphism satisfies, wherever the displayed quotient is initially defined,Consequently, $\eta =({\Phi }^{-1}{)}^{\ast }(dw)$ extends across every apparent denominator zero and across $[0:1:0]$ as a nonzero holomorphic differential form on all of $C$.

One can also check the apparent exceptional points directly. On $z=1$, put $P(X)=4{\prod }_i(X-e_i)$, so $Y^2=P(X)$. At $Y=0$ the roots are distinct, and differentiating giveswhich is regular and nonvanishing there. At $[0:1:0]$, use $u=x/y$, $v=z/y$. The equation is $v=4{\prod }_i(u-e_{i}v)$, so $u$ is a local parameter on a smooth algebraic curve and $v=4u^3+O(u^5)$. Hencealso regular and nonvanishing.

Only distinctness of the $e_i$ is needed. The paper additionally writes $e_i\ne 0$, which is not true for every period lattice: for $\Lambda =\mathbb{Z}+\mathrm{i}\mathbb{Z}$, the symmetry $\wp (\mathrm{i}w)=-\wp (w)$ makes the half-period value at $w=(1+\mathrm{i})/2$ zero, since $\mathrm{i}w\equiv w(\mathrm{mod}\Lambda )$. This unnecessary restriction has no effect on the answer.