The modular lambda function

For $\tau \in \mathfrak{H}$ let ${\mathrm{\Lambda }}_{\tau }:=\mathbb{Z}+\mathbb{Z}\tau$ (Complex lattice and quotient torus) and let ${\mathrm{\wp }}_{{\mathrm{\Lambda }}_{\tau }}$ be its Weierstrass function (Weierstrass p function). Put

$e_1:={\mathrm{\wp }}_{{\mathrm{\Lambda }}_{\tau }}\left(\frac{1}{2}\right),e_2:={\mathrm{\wp }}_{{\mathrm{\Lambda }}_{\tau }}\left(\frac{\tau }{2}\right),e_3:={\mathrm{\wp }}_{{\mathrm{\Lambda }}_{\tau }}\left(\frac{1+\tau }{2}\right),$

the three finite branch values of ${\mathrm{\wp }}_{{\mathrm{\Lambda }}_{\tau }}$ (Degree two of ℘ and its four branch points, clause 3). Those values are pairwise distinct: the discriminant $\mathrm{\Delta }({\mathrm{\Lambda }}_{\tau })=g_2^3-27g_3^2$ is nonzero and $4x^3-g_{2}x-g_3$ has the three distinct roots $e_1,e_2,e_3$ (Nonvanishing of the lattice discriminant). The modular lambda function is

$\lambda (\tau ):=\frac{e_3-e_2}{e_1-e_2}\in \mathbb{C}\setminus \{0,1\}\mathrm{.}$

The value lies in $\mathbb{C}\setminus \{0,1\}$ because $e_1,e_2,e_3$ are pairwise distinct, so numerator $e_3-e_2$ and difference $e_1-e_2$ are nonzero and $e_3-e_2\mathrm{\ne }{e}_1-e_2$. Equivalently, in the cross-ratio convention of The cross-ratio of an ordered quadruple of sphere points,

$\lambda (\tau )=[\mathrm{\infty },e_2;e_1,e_3]=\frac{e_2-e_3}{e_2-e_1},$

matching the displayed formula: the ordered quadruple $(\mathrm{\infty },e_1,e_2,e_3)$ of branch points of the associated Weierstrass cubic determines $\lambda (\tau )$ up to the Möbius transformations fixing $\mathrm{\infty }$. The half-plane conventions are those of The unit disc, the upper half-plane, and Blaschke factors.