Half-period values of the Weierstrass elliptic function ($e_1,e_2,e_3$)

At the three nonzero two-torsion points $z_i$ of $\mathbb{C}/\Lambda$, put $e_i=\wp (z_i)$. The values are distinct because $\wp (z)=\wp (w)$ exactly when $z\equiv \pm w(\mathrm{mod}\Lambda )$.

Let $h_i$ represent a nonzero two-torsion point of a complex torus, and write $e_i=\wp (h_i)$. The product $(\wp (z+h_i)-e_i)(\wp (z)-e_i)$ is an elliptic function whose apparent poles cancel against double zeros. It is therefore constant by Liouville theorem. Its value at $z=0$ is ${\wp }^{\prime \prime }(h_i)/2$. The Weierstrass elliptic differential equation gives ${\wp }^{\prime \prime }(h_i)/2=3e_i^2-g_2/4=(e_i-e_j)(e_i-e_l)$, proving the formula.

At $z=h_i/2$, evenness and periodicity give $\wp (z+h_i)=\wp (z)$. The Weierstrass half-period translation formula therefore identifies $(\wp (h_i/2)-e_i{)}^2$ with $(e_i-e_j)(e_i-e_l)$. Differentiating the Weierstrass half-period translation formula proves the identity, interpreted as equality of meromorphic functions, including the apparent singularities of its quotients.

A two-torsion point satisfies $2z\in \Lambda$. The three nonzero classes are represented by half-periods $\lambda /2$, $\mu /2$, and $(\lambda +\mu )/2$.