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For a period lattice $\Lambda$, define the Weierstrass elliptic function byThe summands on a compact set away from the lattice are $O(|\omega {|}^{-3})$, and that lattice sum converges. Thus the series and its termwise derivative converge normally. The function is even, and ${\wp }^{\prime }=-2{\sum }_{\omega \in \Lambda }(z-\omega {)}^{-3}$ is odd and periodic by reindexing its absolutely convergent series. Hence $\wp (z+{\omega }_j)-\wp (z)$ is constant; evaluation at $-{\omega }_j/2$ and evenness make this constant zero for each basis period. The only poles of wp are double lattice poles, with principal part $(z-\omega {)}^{-2}$.

Let $G_r={\sum }_{\omega \ne 0}{\omega }^{-r}$ for the convergent weights used here. Expanding around zero givesThereforePut $g_2=60G_4$ and $g_3=140G_6$. The elliptic function $({\wp }^{\prime }{)}^2-4{\wp }^3+g_2\wp +g_3$ has a removable singularity at zero and hence at every lattice point. It is entire and periodic, thus bounded on the plane and constant by the Liouville theorem. Its zero constant term makes that constant zero. This proves the Weierstrass elliptic differential equationThe three classes ${\omega }_i/2$, with ${\omega }_3=-{\omega }_1-{\omega }_2$, are the three distinct nonzero two-torsion classes. Oddness and periodicity give ${\wp }^{\prime }({\omega }_i/2)=-{\wp }^{\prime }({\omega }_i/2)$, so they are zeros of the derivative. Their half-period values of the Weierstrass elliptic function $e_i$ are therefore roots of $4X^3-g_{2}X-g_3$.

They are distinct. If $e_i=e_j=e$ with $i\ne j$, the elliptic function $\wp -e$ would have a zero of multiplicity at least two at each of those distinct point classes, since its derivative vanishes there. It has only one pole on the torus, of order two. The argument principle on a fundamental cell equates its total zero and pole orders, contradicting at least four zeros. Thus $e_1,e_2,e_3$ are distinct, and the cubic factors asFor the monic cubic $X^3-(g_2/4)X-g_3/4$, the polynomial discriminant is $(g_2^3-27g_3^2)/16={\prod }_{i<j}(e_i-e_j{)}^2$. Hence the nonvanishing discriminant of a complex lattice isHere $\Delta (\Lambda )$ is the lattice discriminant, not the normalized q-series of Question 1. For $\Lambda =\mathbb{Z}+\mathbb{Z}\tau$, the normalizations give $g_2=(4{\pi }^4/3)E_4$ and $g_3=(8{\pi }^6/27)E_6$, so $\Delta (\Lambda )=(2\pi {)}^{12}\Delta (\tau )$.