An explicit modular equation in two variables for 𝑄(√3)

Abstract

A system of modular equations of norm 2 had been found for the Hilbert modular function field of Q ( 2 ) {\mathbf {Q}}(\sqrt 2 ) in an earlier issue of this journal. Here an analogous system is found for Q ( 3 ) {\mathbf {Q}}(\sqrt 3 ) but with the help of MACSYMA. There are special difficulties in the fact that two spaces of Hilbert modular functions exist for Q ( 3 ) {\mathbf {Q}}(\sqrt 3 ) that can be interchanged by the modular equations. The equations are also a remarkable example of hidden symmetries in the algebraic manifold V 2 {{\mathbf {V}}_2} which is defined in C 4 {{\mathbf {C}}^4} by the modular equation.