= Elliptic collinearity determinant
The determinant of the columns $(1,\wp(z),\wp'(z))$, $(1,\wp(w),\wp'(w))$ and $(1,\wp(-z-w),\wp'(-z-w))$ vanishes identically. Its apparent cubic poles at $z=0,-w$ have order at most two because the coefficient multiplying the cubic-pole entry vanishes linearly. Column coincidences give six zeros, unless $3w$ is a lattice point, when they give four zeros with one multiple zero. In the half-period case, the two apparent zero locations at poles are removable zeros: $\wp'(w)=0$ and the two $1/z$ terms cancel, leaving $O(z)$. Thus the zero count exceeds possible pole order in every case. The <argument principle> proves the identity, which expresses collinearity of the three points on the <Weierstrass equation of an elliptic curve>.
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