Generalized lambda functions and modular function fields of principal congruence subgroups
Given an element \(\tau\) of the complex upper half plane, let \(L_\tau\) be the lattice generated by 1 and \(\tau\), and let \(\wp (z; L_\tau)\) be the Weierstrass \(\wp\)-function associated to \(L_\tau\). If \(e_1\) \(e_2\) and \(e_3\) are 2-division points of the quotient group \(\mathbb C/ L_\tau\), then the elliptic modular lambda function \(\lambda\) is given by \[ \lambda (\tau) = \frac {\wp (e_1; L_\tau) - \wp (e_3; L_\tau)} {\wp (e_2; L_\tau) - \wp (e_3; L_\tau)} . \] In this paper the author proves that the lambda function \(\lambda\) and the modular invariant function \(j\) generate the modular function field with respect to the principal congruence subgroup \(\Gamma (N)\) of \(\mathrm{SL}(2, \mathbb Z)\) with \(N>1\). He also studies its values at at imaginary quadratic points.
scientific article; zbMATH DE number 6350033