The equations for modular function fields of principal congruence subgroups of prime level
For a prime \(p\geq 7\) let \(A(p)\) denote the field of all modular functions on the principal congruence subgroup \(\Gamma(p)\) of level \(p\) in \(\text{SL}_2(\mathbb{Z})\). This field is generated by two functions \(X_2\) and \(X_3\), which are defined by means of so-called Klein forms [\textit{D. Kubert} and \textit{S. Lang}, Math. Ann. 218, 175-189 (1975; Zbl 0311.14005)], and which have zeros and poles only at cusps of \(\Gamma(p)\). The authors show that \(X_3\) is integral over the ring \(\mathbb{Z}[X_2]\), they give detailed information on the irreducible equation \(F_p(X_2,X_3)=0\) of \(X_3\) over \(\mathbb{Z}[X_2]\), and they present an algorithm to compute the polynomial \(F_p(X,Y)\). The result is derived from the behavior of \(X_2\) and \(X_3\) at the cusps of \(\Gamma(p)\), which determines the type of Newton polygon at the places of \(\mathbb{C}(X_2(\tau))\) of an irreducible polynomial of \(X_3(\tau)\) over \(\mathbb{C}(X_2(\tau))\).
principal congruence subgroups of prime level
scientific article; zbMATH DE number 952690
scientific article