Generators and equations for modular function fields of principal congruence subgroups

Abstract

C(X(N)) = C(s, t), FN (s, t) = 0, FN (X,Y ) ∈ Z[ζN ][X,Y ], where FN (X,Y ) is a polynomial of two variablesX and Y such that FN (s, Y ) = 0 is an irreducible equation of t over kN (s). Note that C(X(N)) can be identified with the field A(N) of all the modular functions with respect to Γ (N). Further, the function field kN (X(N)) of X(N) rational over kN is identified with the field FN of all the modular functions of A(N) with kN -rational Fourier coefficients at the cusp i∞. (See §6.2 of Shimura [6].) Thus such generators s and t may be taken from the field FN . The problem considered here is to give such two generators explicitly using Klein forms. Moreover, we would like to know the properties of the polynomial FN (X,Y ). For N prime, this problem was solved by Ishii [2] and by the author and Ishii [1]. In [2], Ishii defined a family of modular functions