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An explicit modular equation in two variables and Hilbert’s twelfth problem. (English) Zbl 0491.12009

The Hilbert modular function field over \(\mathbb Q(\sqrt 2)\) 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 \(\mathbb Q(\sqrt 2)\). Specifically, the form in \(\mathbb Z[\sqrt 2]\), \(x^2+(2+\sqrt 2)2^ty^2\), represents only those primes in the field of \(\mathbb Q(\sqrt 2)\) that split in a tower of fields of singular moduli up to a level depending on \(t\). This has bearing on Hilbert’s twelfth problem, the construction of algebraic numbers by transcendental functions.
Reviewer: Harvey Cohn

MSC:

11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11R37 Class field theory
11R11 Quadratic extensions
11-04 Software, source code, etc. for problems pertaining to number theory
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