## Construction of ray class fields by elliptic units.(English)Zbl 0902.11047

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 $$\omega_1,\omega_2.$$
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.
Reviewer: V.Ennola (Turku)

### MSC:

 11R37 Class field theory 11G16 Elliptic and modular units 11R27 Units and factorization

Zbl 0142.29804
Full Text:

### References:

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