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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)

11R37 Class field theory
11G16 Elliptic and modular units
11R27 Units and factorization
Full Text: DOI EMIS Numdam EuDML
[1] Kubert, D., Lang, S., Modular Units, Grundlehren Math. Wiss., Vol. 244, Springer-Verlag, New-York/ Berlin, (1981). · Zbl 0492.12002
[2] Meyer, C., Die Berechnung der Klassenzahl abelscher Körper über quadratischen Zahlkörpern, Akademie-Verlag, Berlin (1957). · Zbl 0079.06001
[3] Ramachandra, K., Some Applications of Kronecker’s limit formula, Ann. Math.80 (1964), 104-148. · Zbl 0142.29804
[4] Schertz, R., Galoismodulstruktur und elliptische Funktionen, Journal of Number Theory, Vol. 39, No. 3, (1991. · Zbl 0739.11052
[5] Schertz, R., Problèmes de construction en multiplication complexe, Séminaire de Théorie des Nombres de Bordeaux4 (1992), 239-262. · Zbl 0797.11083
[6] Schertz, R., Zur expliziten Berechnung von Ganzheitsbasen in Strahlklassenkörpern über einem imaginär-quadratischen Zahlkörper, Journal of Number Theory, Vol. 34, No. 1 (1990), 41-53. · Zbl 0701.11059
[7] Stark, H., L-functions at s = 1, IV, Advances in Math.35 (1980), 197-235. · Zbl 0475.12018
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