Galois relations for cyclotomic numbers and \(p\)-units
Let \(L/\mathbb{Q}\) be an abelian real field of finite degree and let \(f= f_ L\) denote the conductor of \(L\). For \(\zeta_ f= \exp(2\pi i/f)\), put \(\mathbb{Q}(f)= \mathbb{Q}(\zeta_ f)\). Then the cyclotomic number \(\varepsilon_ L\) attached to \(L\) is defined by \(N_{\mathbb{Q} (f)/L} (1- \zeta_ f)\). In this paper the author studies the relations between the \(\varepsilon_ K\)'s for subfields \(K\) of \(L\), and the relations between the \(\varepsilon_ K\)'s and the cyclotomic \(p\)-units of \(L\) introduced by the author in the preceding paper [Invent. Math. 109, 329--350 (1992; Zbl 0772.11043)]. The first part of the paper studies the cyclotomic numbers of \(L\). Set \({\mathcal S}={\mathcal S}(L)=\{\)subfields \(K\) of \(L\) with \(K\neq \mathbb{Q}\}\cup \{\)prime factors of \(f_ L\}\). For a subset \({\mathcal T}\) of \({\mathcal S}\), let \({\mathcal D}_{\mathcal T}\) denote the \(\mathbb{Z} \text{Gal} (K/\mathbb{Q})\)- submodule of \(L^ \times\) generated by the \(\varepsilon_ K\)'s for a field \(K\in {\mathcal T}\) and an integer \(r\in{\mathcal T}\). Then \({\mathcal D}_{\mathcal T}\) is called the group of \({\mathcal T}\)-cyclotomic numbers of \(L\). Set \(W_{\mathcal T}= \oplus_{K\in{\mathcal T}} \mathbb{Z} \text{Gal} (K/\mathbb{Q}) \oplus_{r\in{\mathcal T}} \mathbb{Z}\), and define a surjective homomorphism \[ \alpha_{\mathcal T}: W_{\mathcal T}\to {\mathcal D}_{\mathcal T} \qquad \text{by} \qquad \alpha_{\mathcal T}(\sum w_ K\oplus \sum w_ r)= \prod \varepsilon_ K^{w_ K}\times \prod r^{w_ r}. \] Let \(R_{\mathcal T}\) denote the kernel of \(\alpha_{\mathcal T}\). For \(\sigma,\sigma'\in{\mathcal T}\), let \(w(\sigma/ \sigma')\) denote an element of \(W_{\mathcal T}\) defined explicitly using the norm and Frobenius symbols, and let \(U_{\mathcal T}\) denote the \(\mathbb{Z} \text{Gal} (L/\mathbb{Q})\)- submodule generated by \(w(\sigma/\sigma')\) for \(\sigma, \sigma'\in {\mathcal T}\). Then it is seen that \(U_{\mathcal T}\subset R_{\mathcal T}\). Let \({\mathcal G}= {\mathcal G}(L)=\{\)cyclic subfields \(K\) of \(L\) with \(K\neq\mathbb{Q}\}\cup\{\)prime factors of \(f_ L\}\). Then the author proves as the main result of the paper that \(U_{\mathcal G}= R_{\mathcal G}\) and \(R_{\mathcal T}/ U_{\mathcal T}\) is finite if \({\mathcal T}\supset{\mathcal G}\). In the second part of the paper the author uses the cyclotomic numbers to study the cyclotomic \(p\)-units. Let \(p\) be a fixed rational prime and assume that \(p\) is completely decomposed in \(L\). Let \(E_ p^ +(L)\) denote the group of totally real \(p\)-units of \(L\). Then for \(\sigma\in{\mathcal S}\) the cyclotomic \(p\)-unit \(\kappa_ \sigma\) is defined as an element of \(E_p^+(L)\oplus_{\mathbb{Z}} \mathbb{Z}_ p\), whose properties were studied in the paper mentioned above. Let \({\mathcal K}_{\mathcal S}\) denote the group of cyclotomic \(p\)-units of \(L\), and define a surjective homomorphism \(\delta_{\mathcal S}: W_{\mathcal S}\to {\mathcal K}_{\mathcal S}\) similarly to the cyclotomic numbers. Then it is proved that there exists a \(\mathbb{Z} \text{Gal} (L/\mathbb{Q})\)-isomorphism \(\theta: {\mathcal D}_{\mathcal S}\to {\mathcal K}_{\mathcal S}\) such that \(\theta\circ \alpha_{\mathcal S}= \delta_{\mathcal S}\).
scientific article; zbMATH DE number 556363