On algebraic values of function exp (2ni x + log log y)

TL;DR

This paper proves that for most square-free integers, the exponential of a specific transcendental function with algebraic arguments in quadratic fields yields algebraic numbers that generate the Hilbert class field of imaginary quadratic fields.

Contribution

It establishes a new link between transcendental functions and algebraic number fields, specifically showing how certain exponential values generate Hilbert class fields.

Findings

For all but finitely many square-free integers d, the function value is algebraic.

The function value generates the Hilbert class field of the imaginary quadratic field with discriminant -d.

The result applies to algebraic arguments in real quadratic fields.

Abstract

It is proved that for all but a finite set of the square-free integers $d$ the value of transcendental function $\exp~(2\pi i ~x+\log\log y)$ is an algebraic number for the algebraic arguments $x$ and $y$ lying in a real quadratic field of discriminant $d$. Such a value generates the Hilbert class field of the imaginary quadratic field of discriminant $-d$.