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

Cohn, Harvey; Deutsch, Jesse

doi:10.1090/S0025-5718-1988-0929553-4

An explicit modular equation in two variables for $\textbf {Q}(\sqrt 3)$
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Math. Comp. 50 (1988), 557-568 Request permission

Abstract:

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

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