An explicit modular equation in two variables and Hilbert’s twelfth problem

Abstract:

The Hilbert modular function field over ${\mathbf {Q}}(\surd 2)$ has generators satisfying modular equations when the arguments are multiplied by factors of norm two. These equations are found by machine use of Fourier series and are further used to show computationally that Weber’s ring class field theory for rationals has an illustration of Hecke’s type for ${\mathbf {Q}}(\surd 2)$. This has bearing on Hubert’s twelfth problem, the generation of algebraic fields by transcendental functions.