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


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