## On the density of Abelian number fields.(English)Zbl 0566.12001

The problem is to determine J(d,G) (resp. N(X,G)), the number of Abelian number fields K with discriminants $$d(K)=d$$ (resp. $$| d(K)| \leq K)$$ and Galois groups Gal(K/$${\mathbb{Q}})\cong G$$. H. Hasse and H. Cohn studied it for $$G\cong C(3)$$ (C(m) denotes the cyclic group of order m). A. M. Baily solved it when $$G\cong C(4)$$, C(2)$$\times C(2)$$, and C(2)$$\times C(2)\times C(2)$$ [J. Reine Angew. Math. 315, 190-210 (1980; Zbl 0421.12007); ibid. 328, 33-38 (1981; Zbl 0503.12010)] (which contain slight coefficient mistakes as indicated in the reviewer’s following papers). The reviewer solved this problem when $$G\cong C(p)\times C(p)\times...\times C(p)$$ (n factors) for general n and prime p [Sci. Sin., Ser. A 27, 345-351, 1018-1026 (1984; Zbl 0538.12006 and Zbl 0565.12006)]. In particular, $J(d,G)=p^{-n(n-1)/2}\cdot \sum^{n- 1}_{i=0}(-1)^ i p^{i(i-1)/2} t_{n-i}^{\omega (d)}/(\pi_ i\pi_{n-i}),$
$N(X,G)=C X^{p/((p-1)| G|)} (\log X)^{(p^ n-p)/(p-1)}+O(R),$ where $$t_ i=p^ i-1$$, $$\pi_ i=t_ 1t_ 2...t_ i$$, $$\omega$$ (d) is the number of prime divisors of d, C an explicitly given constant, $$R=X^{p/((p-1)| G|)} (\log X)^{(p^ n-2p+1)/(p-1)}$$; except that the formula of J(d,G) alters slightly when $$p=2$$ and $$2^{3\cdot 2^{n-1}}| d$$. This problem has been investigated by B. M. Urazbaev and many other Soviet mathematicians in several cases for $$G\cong C(p^ v)$$, C(p)$$\times...\times C(p)$$, and $$C(p_ 1)\times C(p_ 2)\times C(p_ 3)$$ under the restriction $$(| G|,d)=1.$$
The present work is a study of the general case with G any finite Abelian group. A general complicated formula for J(d,G) is obtained, and for any $$\epsilon >0$$, $N(X,G)=X^{p_ 1/((p_ 1-1)| G|)} P(\log X)+O(X^{r+\epsilon}),$ where $$p_ 1$$ is the least prime divisor of $$| G|$$, P is a polynomial of degree $$(p^ h_ 1-p_ 1)/(p_ 1-1)$$ if $$| G| \not\equiv 2 mod 4$$, or else P(log X) is a constant, h is the number of cyclic factors in the representation of the $$p_ 1$$-Sylow subgroup of G as a direct product of cyclic groups, $$0<r<p_ 1/((p_ 1-1)| G|)$$. Most of the earlier results can be deduced from these results. When $$G\cong C(p)\times...\times C(p)$$, these lead to formulae in accordance with those mentioned above except that the constant C here is in more complicated form and the error terms differ because of the substitution of P(log X) for log X in the main term.
Other special cases are: $$J(d,C(p^ v))=p^{-v+1} (p-1)^{- 1}\prod^{\omega (d)}_{j=1}(p^{b_ j-1} (p-1))$$ with $$b_ j\in \{1,2,...,v\}$$ and $$q_ j\equiv 1 mod p^{b_ j}$$ when $$q_ j\neq p$$ $$(d=q_ 1^{e_ 1}...q_{\omega}^{e_{\omega}})$$ [J. Värmon obtained this in his thesis (Lund, 1925)], $$J(d,C(p_ 1)\times...\times C(p_ t))=\prod^{t}_{i=1}(p_ i-1)^{T_ i},$$ $$0\leq T_ i\leq \omega (d)-1$$. The main interest lies in the combinatorial aspect of counting character groups $$\cong G$$ to obtain J(d,G), the analytical machinary in obtaining N(X,G) is standard.
Reviewer: X.Zhang

### MSC:

 11R18 Cyclotomic extensions 11R45 Density theorems 11R23 Iwasawa theory

### Keywords:

discriminant density; Abelian number fields; discriminants

### Citations:

Zbl 0421.12007; Zbl 0503.12010; Zbl 0538.12006; Zbl 0565.12006