Brauer-Kuroda relations for \(S\)-class numbers (Q2717619)

In this article, the author describes a systematic method for deriving explicit class number formulas from the Brauer-Kuroda relations associated to character relations. NEWLINENEWLINENEWLINELet \(K\) be a number field, \(S\) a finite set of primes in \(K\), and \(G\) a finite group of field automorphisms acting on \(K\). Let \(X\) be \(G\)-set, i.e., a finite set on which \(G\) acts (thus \(S\) is a \(G\)-set); then the set \(K_X\) of \(G\)-equivariant maps \(X \rightarrow K\) can be identified with a product of number fields. Let \(h(K_X)\) (\(R(K_X)\); \(U(K_X)\); \(w(K_X)\)) denote the product of the \(S\)-class numbers (\(S\)-regulators; \(S\)-unit groups; number of roots of unity) of these fields. Let \(\pi_X: G \rightarrow \mathbb Z\) denote the map sending \(g \in G\) to the number of fixed points of \(g\) in \(X\). Two \(G\)-sets \(X\), \(Y\) are called linearly equivalent if \(\pi_X = \pi_Y\); such linear equivalences encode character relations, and in fact the Brauer-Kuroda relations can be expressed in the form NEWLINE\[NEWLINE \frac{h(K_X) R(K_X)}{w(K_X)} = \frac{h(K_Y) R(K_Y)}{w(K_Y)}. NEWLINE\]NEWLINE NEWLINENEWLINENEWLINENow let \(\mathbb Z X\) denote the free \(\mathbb Z\)-module on \(X\) with \(G\) acting through permuting \(X\). If \(\pi_X = \pi_Y\), then there are injective \(\mathbb Z G\)-linear homomorphisms \(\phi: \mathbb Z X \rightarrow \mathbb Z Y\), and they induce group homomorphisms \(\phi_U: U(K_Y) \rightarrow U(K_X)\) with finite kernel and cokernel. NEWLINENEWLINENEWLINEDefine \(B = (\prod (\text{coker} \phi)^{D({\mathfrak p})}:(\text{coker} \phi)^G)\), where the product is over a set of representatives of the \(G\)-orbits of \({\mathfrak p}\) in \(S\), and where \(D({\mathfrak p})\) is the stabilizer of \({\mathfrak p}\) in \(G\). Then the main result of this article is the divisiblity relation NEWLINE\[NEWLINE \frac{h(K_Y)}{h(K_X)} \;\Big|\;B \cdot \frac{w(K_Y)}{w(K_X)} NEWLINE\]NEWLINE for \(S\)-class numbers, where \(x \mid y\) means \(y/x \in \mathbb Z\). NEWLINENEWLINENEWLINEAs an application, an explicit class number formula for Galois extensions \(K/F\) with Galois group \(G \simeq (\mathbb Z/p\mathbb Z)^m\) is derived, and it is shown that the ratio \(h/h'\) of the class numbers of \(K = \mathbb Q(a^{1/8})\) and \(K' = \mathbb Q((16a)^{1/8})\) assumes only the values \(1/2\), \(1\), and \(2\).