Distribution of discriminants of abelian extensions. (English) Zbl 0628.12006

Wright, David J.

Proc. Lond. Math. Soc., III. Ser. 58, No. 1, 17-50 (1989).

In this paper, class field theory is used to determine the distribution of the relative discriminants of all Galois extensions of a given global field k with Galois group isomorphic to a given finite abelian group G. In the case that the base field has characteristic \(p>0\), the results established assume that p does not divide the order of G. The analogous problem when only those extensions satisfying given localization conditions at finitely many places of k are counted is also solved, given the existence theorem of Grunwald, Wang, and Hasse.
The principle technique involves locating the rightmost pole of the Dirichlet series formed by summing over all these extensions the absolute norms of the relative discriminants raised to the -s power. This rightmost pole is identified by rewriting the Dirichlet series as a linear combination of Euler products which may be compared with products of finitely many Dedekind zeta-functions.

MSC:

11R18	Cyclotomic extensions
11R23	Iwasawa theory
11R58	Arithmetic theory of algebraic function fields
11R37	Class field theory
11R42	Zeta functions and \(L\)-functions of number fields

Keywords:

class field theory; distribution of discriminants; Dirichlet series; relative discriminants; Euler products; abelian extensions