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Ray class invariants over imaginary quadratic fields. (English) Zbl 1279.11060

Summary: Let \(K\) be an imaginary quadratic field of discriminant less than or equal to \(-7\) and \(K_{(N)}\) be its ray class field modulo \(N\) for an integer \(N\) greater than 1. We prove that the singular values of certain Siegel functions generate \(K_{(N)}\) over \(K\) by extending the idea of our previous work [Bull. Lond. Math. Soc. 41, No. 5, 935–942 (2009; Zbl 1188.11031)]. These generators are not only the simplest ones conjectured by R. Schertz [J. Théor. Nombres Bordx. 9, 383–394 (1997; Zbl 0902.11047)], but also quite useful in the matter of computation of class polynomials. We indeed give an algorithm to find all conjugates of such generators by virtue of the works of A. Gee [J. Théor. Nombres Bordx. 11, No. 1, 45–72 (1999; Zbl 0957.11048)] and P. Stevenhagen [Adv. Stud. Pure Math. 30, 161–176 (2001; Zbl 1097.11535)].

MSC:

11G16 Elliptic and modular units
11F11 Holomorphic modular forms of integral weight
11F20 Dedekind eta function, Dedekind sums
11G15 Complex multiplication and moduli of abelian varieties
11R37 Class field theory
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