On Brauer-Kuroda type relations of \(S\)-class numbers in dihedral extensions (Q2904016)

Let \(F/k\) be a Galois extension of number fields, and assume that its Galois group is the dihedral group \(D_{2q}\) of order \(2q\), where \(q\) is an odd integer. Let \(S\) be a finite set of primes in \(F\), and assume that \(S\) is fixed by the action of the Galois group and that it contains all the archimedean primes. Let \({\mathcal O} = {\mathcal O}_S\) denote the ring of \(S\)-integers, i.e., the ring of all algebraic numbers in \(F\) which are integral at all places not in \(S\), let \(C_S(F)\) denote the \(S\)-class group of \(F\), which is, by definition, the class group of the Dedekind ring \({\mathcal O}\) (reviewer's remark: this definition excludes the \(S\)-class groups for sets \(S\) not necessarily containing all Archimedean primes, in particular, it excludes the class groups in the struct sense), and let \(h_S(F) = \# C_S(F)\) denote the \(S\)-class number.NEWLINENEWLINEThe author's main result is the following generalization of a class number formula for \(D_{2q}\)-extensions obtained by \textit{L. Caputo} and \textit{F. Nuccio} [``On fake \(\mathbb Z_p\) extensions of number fields'', \url{arXiv:0807.1135}]: Let \(p\) be an odd prime number, \(F/k\) be a \(D_{2p}\)-extension of number fields, let \(K/K\) be the quadratic subextension inside \(F/k\), and let \(L\) and \(L'\) be distinct intermediate extensions of degree \(p\) over \(k\). As above, let \({\mathcal O}_F\) denote the ring of \(S\)-integers in \(F\), and \({\mathcal O}_F^\times\) its unit group. Let \(r_s\) denote the rank of the group of \(S\)-units, and let \(a(F/K,S)\) denote the number of primes of \(k\) that lie below those in \(S\) and have decomposition group \(D_{2p}\). Finally, set \(\delta = 3\) or \(1\) according as \(L/K\) is generated by a \(p\)-th root of a non-torsion \(S\)-unit or not. Then NEWLINE\[NEWLINE \frac{h_S(F) h_S(k)^2}{h_S(K) h_S(L)} = p^{\alpha/2} \cdot [{\mathcal O}_F^\times : {\mathcal O}_L^\times{\mathcal O}_{L'}^\times {\mathcal O}_K^\times], NEWLINE\]NEWLINE where NEWLINE\[NEWLINE \alpha = 2r_S(k) - r_S(K) - \frac{r_S(F) - r_S(K)}{p-1} + a(F/K,S) - \delta.NEWLINE\]NEWLINE The proof is mainly representation theoretic and uses the ``regulator constants'' introduced in [\textit{T. Dokchitser} and \textit{V. Dokchitser}, Ann. Math. (2) 172, No. 1, 567--596 (2010; Zbl 1223.11079)], which were called ``Dokchitser constants'' by the author in [Math. Proc. Camb. Philos. Soc. 148, No. 1, 73--86 (2010; Zbl 1242.11038)].