Construction of ray class fields by elliptic units
Let \(K\) be an imaginary quadratic field, and let \(K(\mathfrak f)\) be a ray class field over \(K\), where the conductor \(\mathfrak f\) is an integral ideal of the ring of integers of \(K\). Let \(\omega_1,\omega_2\) be a \(\mathbb Z\)-basis of \(\mathfrak f\) with \(\text{Im} (\omega_1/\omega_2)>0\). The normalized Klein form is defined by \[ \varphi(z)=2\pi i\exp(-zz^\ast/2)\sigma(z\mid\mathfrak f) \eta(\omega_1/\omega_2)^2\omega_2^{-1}, \] where \(\sigma\) denotes the \(\sigma\)-function of \(\mathfrak f\), \(\eta\) is the Dedekind \(\eta\)-function, and \(z^\ast\) is defined by \(z^\ast=z_1\eta_1+ z_2\eta_2\) with the real coordinates \(z_1,z_2\) of \(z=z_1\omega_1+z_2\omega_2\) and the quasi-periods \(\eta_1,\eta_2\) of the elliptic Weierstrass \(\zeta\)-function of \(\mathfrak f\) belonging to \(\omega_1,\omega_2.\) \textit{K. Ramachandra} [Ann. Math. (2) 80, 104-148 (1964; Zbl 0142.29804)] has proved that \(K(\mathfrak f)\) can be generated by a complicated product involving high powers of singular values of \(\varphi\) and singular values of the discriminant. The author's aim is to show that in many cases it is sufficient to take a power of one singular value of \(\varphi\) or a quotient of two such values. Numerical examples indicate that the coefficients of the minimal polynomials of the constructed numbers are rather small.
scientific article; zbMATH DE number 1164094
scientific article