×

zbMATH — the first resource for mathematics

On the distribution of norms of ideals in given ray-classes and the theory of central ray-class fields. (English) Zbl 0694.12007
Let \(K\) be an algebraic number field of finite degree over the rationals. \(\mathbb{Z}_K\) will denote the ring of integers in \(K\). Let \(\mathfrak q\) be some conductor in \(K\), containing as factors all the real infinite places of \(K\) (each to the first power). The range \(R(n)\) of \(n\in\mathbb{N}\) is defined by
\[ R(n):=\{[\mathfrak a];\ N\mathfrak a=n\}. \]
Here, \([\mathfrak a]\) denotes the ray class of \(\mathfrak a \bmod \mathfrak q\) and \(\mathfrak a\) runs over all integral ideals of \(K\) satisfying \(N\mathfrak a=n\), \(\mathfrak a+\mathfrak q'=\mathbb{Z}_K\), where \(\mathfrak q'\) is the “finite part” of \(\mathfrak q\).
The author gives a detailed analysis of the variation of \(R(n)\) with \(n\), making extensive use of Dirichlet series and the Chebotarev density theorem in the classical global theory of class fields. In particular he obtains an “equidistribution” theorem for norms of ideals in given ray classes, together with a proof that “almost all” norms have a maximal range.
The investigations improve previous results of the author [Acta Arith. 33, 53–63 (1977; Zbl 0363.10025), Algebr. Number Fields, Proc. Symp. Lond. Math. Soc. 1975, 485–495 (1977; Zbl 0357.12015)].

MSC:
11R37 Class field theory
11R45 Density theorems
PDF BibTeX Cite
Full Text: DOI EuDML