Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 23G / Solution

An elliptic function is meromorphic and $\Lambda$-periodic. The argument principle on opposite sides of a period parallelogram cancels, so zeros and poles have equal total multiplicity. The Weierstrass elliptic functionis elliptic. Its half-period values $e_i$ are distinct, and comparing poles and leading coefficients proves ${\wp }^{\prime 2}=4{\prod }_i(\wp -e_i)$. The half-period addition formula givesDifferentiate and use the same formula at $z={\omega }_1/4$ to identify $(e_1-e_2)(e_1-e_3)=(\wp ({\omega }_1/4)-e_1{)}^2$, yielding the required derivative ratio.