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Choose an oriented basis ${\omega }_1,{\omega }_2$ of the period lattice, so $\mathrm{Im}({\omega }_2/{\omega }_1)>0$. The Weierstrass elliptic function is constructed by the normally convergent seriesOn compact sets away from the lattice the summands are $O(|\omega {|}^{-3})$, which is summable over a plane lattice. It is even, has double poles at the lattice with zero residues, and is periodic. To check periodicity without splitting divergent series, differentiate termwise: ${\wp }^{\prime }(z)=-2{\sum }_{\omega \in \Lambda }(z-\omega {)}^{-3}$ is invariant under lattice shifts. Hence $\wp (z+{\omega }_j)-\wp (z)$ is constant; evenness evaluated at $z=-{\omega }_j/2$ makes this constant zero.

The Weierstrass sigma function is the canonical productIts logarithmic tails are likewise $O(|\omega {|}^{-3})$, so it defines an entire function with simple zeros exactly at the lattice and ${\sigma }^{\prime }(0)=1$. Pairing opposite lattice factors makes it odd. Its logarithmic derivative is the Weierstrass zeta functionThus ${\eta }_j=\zeta (z+{\omega }_j)-\zeta (z)$ is constant. Integrating this identity and evaluating the sigma ratio at $z=-{\omega }_j/2$ givesIntegration of zeta around a fundamental cell, containing one pole of residue one, gives the Legendre relation for Weierstrass quasi-periods ${\eta }_1{\omega }_2-{\eta }_2{\omega }_1=2\pi i$. These formulas explain why sigma itself is not an elliptic function, but balanced products of translates can be.

For completeness, the basic algebraic property of wp is $({\wp }^{\prime }{)}^2=4{\wp }^3-g_2\wp -g_3$, where $g_2=60{\sum }_{\omega \ne 0}{\omega }^{-4}$ and $g_3=140{\sum }_{\omega \ne 0}{\omega }^{-6}$. The identity follows by canceling the principal parts using the Laurent expansion: the difference is an entire elliptic function, hence constant, and its constant term is zero. The discriminant is nonzero; equivalently the three half-period values are distinct. These properties lead to the cubic embedding in 2(ii).

Now let $F$ be a nonzero meromorphic function on the torus and lift it to a periodic meromorphic function on the plane. Its logarithmic derivative $g=F^{\prime }/F$ is periodic. Choose a fundamental parallelogram with no zero or pole on its boundary. The residue theorem applied to $g$ gives equality of the total zero and pole multiplicities. Applied to $zg$, it gives the difference of the sums of the zero and pole lifts. Pairing opposite edges explicitly yieldsEach integral of $F^{\prime }/F$ on the right is $2\pi i$ times an integer, since its endpoint values of $F$ coincide and the image path avoids zero. Therefore the zero sum and pole sum agree in $\mathbb{C}/\Lambda$.

Conversely, suppose the point sums agree modulo the lattice. Change one chosen lift of a point by a lattice element so the two sums agree exactly in $\mathbb{C}$; this changes no point of the torus. Thenhas the prescribed divisor. Its multiplier under either basis period is $\exp ({\eta }_j(\sum Q_i-\sum P_i))=1$, so it descends to the torus. Repetitions contribute their appropriate multiplicities. This proves the principality criterion on a complex elliptic curve:There is an implicit qualification in the wording about separate zero and pole divisors: their supports must be disjoint. If a point occurs on both lists, cancel common multiplicities first when prescribing the divisor difference. A function cannot simultaneously have a zero and a pole at the same point; for example identical one-point lists satisfy the sum condition but do not specify separate zero and pole divisors. For $n=0$, any nonzero constant supplies the empty divisor.

The Weierstrass elliptic function and its derivative give the maponto the Weierstrass equation of an elliptic curve $Y^{2}Z=4X^3-g_{2}X{Z}^2-g_3{Z}^3$. Near zero, multiplying these coordinates by $z^3$ gives $[z+O(z^5):-2+O(z^4):z^3]$, so the map extends holomorphically and is an immersion at the origin.

The meromorphic map $\wp :E\to {\mathbb{P}}^1$ has degree two, since its only pole on the torus has order two. Its evenness implies $\wp (u)=\wp (v)$ exactly when $u\equiv v$ or $u\equiv -v$. Oddness of ${\wp }^{\prime }$ distinguishes these two alternatives unless they are the same point class. At each nonzero half-period, ${\wp }^{\prime }=0$. These three points exhaust its zeros, since its only pole has order three. The three half-period values are distinct: equality of two of them would give two distinct zeros each of order at least two for $\wp -e$, contradicting its total pole order two. They are roots of $4X^3-g_{2}X-g_3$, so the polynomial has distinct roots and the cubic is smooth. At a half-period $h$ with value $e$, ${\wp }^{\prime \prime }(h)=2{\prod }_{e^{\prime }\ne e}(e-e^{\prime })\ne 0$. Elsewhere ${\wp }^{\prime }$ is nonzero. The coordinate pair therefore has nonzero derivative at every finite point, and the map is injective. Compactness and the smooth cubic equation then identify it as an embedding onto the cubic.

The hyperplane section at infinity has divisor $3[O]$. Thus the intersection divisor $D_{\ell }$ of any line with this cubic is linearly equivalent to $3[O]$. Applying the principality criterion on a complex elliptic curve to $D_{\ell }-3[O]$ shows that its three points, counted with intersection multiplicity, have sum zero in $E$.

Conversely, if the degree-three divisor $D=[u]+[v]+[w]$ has point sum zero, Question 1 gives a meromorphic function with divisor $D-3[O]$. A meromorphic function with no poles away from $O$ and pole order at most three lies inTo prove this description, subtract multiples of ${\wp }^{\prime }$ and $\wp$ to cancel the cubic and quadratic Laurent principal parts. The remaining possible simple-pole coefficient is a residue, which must be zero because it is the only pole on a compact torus. The remaining holomorphic elliptic function is constant. Thus the constructed function is a linear form in the projective coordinates, and its line has intersection divisor exactly $D$. This proves the chord-and-tangent collinearity criterion is $u+v+w=0$ in $E$, with multiplicities understood.

For three pairwise distinct point classes away from $O$, ordinary collinearity is equivalent to linear dependence of the coordinate columns, soThe original PDF has this three-column determinant; the TeX's extra copied column is a transcription defect.

There is also a genuine implicit distinctness condition in the printed determinant assertion. If $u=v={\omega }_1/10$ and $w={\omega }_1/5$, all entries are finite and the determinant is zero because two columns coincide, but $u+v+w=2{\omega }_1/5\notin \Lambda$. Thus the unrestricted literal equivalence is false. To specify repeated intersections, use the confluent elliptic collinearity determinant: for a double point at $u$ replace the second column by $c^{\prime }(u)$, where $c(z)=(1,\wp (z),{\wp }^{\prime }(z){)}^t$; for a triple point use $c(u),c^{\prime }(u),c^{\prime \prime }(u)$. These impose tangency or inflection, and recover the correct conditions $2u+w\in \Lambda$ or $3u\in \Lambda$. If a point is $O$, use the homogeneous coordinates or a local homogeneous lift, since the displayed affine entries have poles there.