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An elliptic resolvent. (English) Zbl 0953.11034

The main subject of this paper is a generalization of previous results R. Schertz [J. Number Theory 39, 285-326 (1991; Zbl 0739.11052)]. Let \(K\) be an imaginary quadratic number field and \({a}\), \( {b}\) integer ideals in \(K\). The author gives an explicit Galois generator for the ring of integers of the ray class field over \(K\), in relative extensions \(K( {qa})/K( {q})\) with some hypothesis on \( {q}\) and \( {a}\). Starting with new resolvent relations and an analytical interpretation for Galois extensions of fields of elliptic functions with respect to lattices \(\Gamma\subset\hat \Gamma\), he obtains the main result by using singular values of these functions in the complex multiplication context. Then the author applies his result to give explicit generators for Kummer orders considered in A. Srivastav and M. J. Taylor [Invent. Math. 99, 165-184 (1990; Zbl 0705.14031)].

MSC:

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R27 Units and factorization
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References:

[1] W. Bley, R. Boltje, Relative Lubin-Tate formal groups and module structure over Hopf orders, 1997; W. Bley, R. Boltje, Relative Lubin-Tate formal groups and module structure over Hopf orders, 1997 · Zbl 0979.11053
[2] P. Cassou-Noguès, A. Jehanne, Espaces homogènes principaux et points de 2-division de courbes elliptiques, 1998; P. Cassou-Noguès, A. Jehanne, Espaces homogènes principaux et points de 2-division de courbes elliptiques, 1998
[3] Cassou-Noguès, P.; Taylor, M. J., Progress in Mathematics, 66 (1987)
[4] Schertz, R., Konstruktion von Potenzganzheitsbasen in Strahlklassenkörpern über imaginär-quadratischen Zahlkörpern, J. Reine Angew. Math., 398, 105-129 (1989) · Zbl 0666.12006
[5] Schertz, R., Zur expliziten Berechnung von Ganzheitsbasen in Strahlklassenkörpern über einem imaginär-quadratischen Zahlkörper, J. Number Theory, 34 (1990) · Zbl 0701.11059
[6] Schertz, R., Galoismodulstruktur und elliptsche Funktionen, J. Number Theory, 39, 285-326 (1991) · Zbl 0739.11052
[7] Silverman, J., The arithmetic of elliptic curves, Grad. Texts in Math. (1986), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0585.14026
[8] Srivastav, A.; Taylor, M. J., Elliptic curves with complex multiplication and Galois module structure, Invent. Math., 99, 165-184 (1990) · Zbl 0705.14031
[9] Taylor, M. J., Mordell-Weil groups and the Galois module structure of rings of integers, Illinois J. Math., 32, 428-452 (1988) · Zbl 0631.14033
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