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An elliptic function is a meromorphic $f:\mathbb{C}\to \hat{\mathbb{C}}$ with two real-linearly independent periods ${\omega }_1,{\omega }_2$. A fundamental cell may be taken as the half-open parallelogramwith $z_0$ chosen so that its boundary contains no zero or pole under discussion. The argument principle and cancellation on opposite sides show that a nonconstant elliptic function has the same number of zeros and poles in a cell, counting multiplicity.

The Weierstrass elliptic function satisfiesWriting $G_{2r}={\sum }_{\omega \in \Lambda \setminus \{0\}}{\omega }^{-2r}$, the Laurent coefficients of the Weierstrass elliptic function giveThus $g_2=60G_4$ and $g_3=140G_6$ cancel every nonremovable term in the Laurent series of ${\wp }^{\prime 2}-4{\wp }^3+g_2\wp +g_3$ at each point of the period lattice. The resulting elliptic function is an entire function and is bounded, hence is zero by Liouville theorem.

For a period lattice $\Lambda \subset \mathbb{C}$, the Weierstrass elliptic function is the normally convergent lattice sumIt is an even elliptic function with a double pole of principal part $(z-\omega {)}^{-2}$ and zero residue at every point $\omega$ of the period lattice.

The Weierstrass elliptic function and its derivative give an isomorphism $z\mapsto (\wp (z),{\wp }^{\prime }(z))$ from a one-dimensional complex torus to a smooth plane cubic, with $0$ sent to its identity at infinity. The three nonzero two-torsion points of a complex torus are zeros of ${\wp }^{\prime }$. Its triple pole implies that these three zeros are simple. If two half-period values of the Weierstrass elliptic function coincided, $\wp -e$ would have at least four zeros counted with multiplicity, contradicting its double pole. The three cubic roots are therefore distinct, proving smoothness of an algebraic variety. The degree-two map $\wp$ has fibers $\{z,-z\}$, distinguished by ${\wp }^{\prime }$ away from the half-periods, proving bijectivity. A line pulls back to an elliptic function with a triple pole, and the zero-pole sum of an elliptic function says that its three intersections sum to zero on the torus. This gives exactly the chord-and-tangent group law, including repeated intersections interpreted by intersection multiplicity.

For a nonzero elliptic function with period lattice $\Lambda$, its zeros of a function $a_j$ and poles $b_j$, repeated with their multiplicities, have equal total number and satisfy ${\sum }_j{a}_j-{\sum }_j{b}_j\in \Lambda$. Choose a fundamental parallelogram avoiding all zeros and poles on its boundary. The argument principle applied to $f^{\prime }/f$ gives equality of the numbers. Applying the residue theorem to $z{f}^{\prime }/f$ gives their difference of sums. Pairing opposite edges, the extra factors are the two lattice generators multiplied by integrals of $f^{\prime }/f$ along an edge. These integrals are integer multiples of $2\pi i$, because the endpoints have the same nonzero function value. Thus the difference of sums is a lattice element. This links zeros of a pulled-back line to the chord-and-tangent group law.