The conductor density of abelian number fields. (English) Zbl 0727.11041

Mäki, Sirpa

J. Lond. Math. Soc., II. Ser. 47, No. 1, 18-30 (1993).

Let G be a finite abelian group, and let K range over all absolutely abelian number fields with Galois group Gal(K/\({\mathbb{Q}})\cong G\). Let \({\mathcal N}(x)=\sum_{0<f\leq x}N(f)\), where N(f) is the number of fields K with conductor equal to f. The main results are: (i) An explicit formula for N(f). (ii) An asymptotic formula of the type \({\mathcal N}(x)=x P(\log x)+O(x^{1-\delta +\epsilon}),\) where P is a polynomial and \(\delta\) is a given positive constant. Explicit expressions for the degree and the leading coefficient of P are computed. The results should be compared with those of M. J. Taylor [J. Lond. Math. Soc., II. Ser. 29, 211- 223 (1984; Zbl 0535.12009)] who considered the case when G is cyclic and the basefield contains certain roots of unity.

Reviewer: S.Mäki (Turku)

MSC:

11R20	Other abelian and metabelian extensions
11N45	Asymptotic results on counting functions for algebraic and topological structures

Keywords:

conductor density; abelian number fields; explicit formula; asymptotic formula

Citations:

Zbl 0535.12009