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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.

Put $\mathcal{D}={\mathcal{K}}_X^{\ast }/{\mathcal{O}}_X^{\ast }$, the quotient sheaf of abelian groups. At a closed point $P$, the discrete valuation gives an isomorphismIndeed, a local parameter on a smooth algebraic curve has valuation one, and the valuation kernel is precisely the unit group of the discrete valuation ring. Thus a local section represented by $f$ determines its local principal divisor, and changing $f$ by a regular function which is a unit does not change those coefficients.

These local valuations identify $\Gamma (X,\mathcal{D})$ with $\mathrm{Div}(X)$. To see that they give finite support, represent a section locally by finitely many rational functions on an algebraic variety, using Noetherian Zariski topology to take a finite cover. Each representative has only finitely many zeros and poles, so their combined support is finite. Conversely, for $D=\sum n_{P}P$, choose at each support point a local parameter $t_P$ and shrink its neighbourhood to exclude every other zero or pole of $t_P$ and every other support point. The local rational equation $t_P^{n_P}$ represents the desired valuations there; on the complement of the support use $1$. On overlaps these equations differ by units, so their classes glue in the quotient sheaf. Finally, a section with every valuation zero has zero germ everywhere and is the identity section. Hence the identification is both surjective and injective.

The map on global sections sends $f\in k(X{)}^{\ast }$ to $\mathrm{div}(f)$. Therefore its cokernel is

In a regular local ring of dimension $d$, a regular system of parameters is a list in the maximal ideal whose images form a residue-field basis of $\mathfrak{m}/{\mathfrak{m}}^2$. The Nakayama lemma shows that it generates the maximal ideal. At a smooth closed point over an algebraically closed field, these are the local parameters in all dimensions, extending the single local parameter on a smooth algebraic curve.