Formules de classes pour les corps abéliens réels. (Class formulae for real Abelian fields) (Q5944298)

Let \(K\) be a real Abelian number field of degree \(n\) and Galois group \(G\) over \(\mathbb Q\). Let \(h_K\), \(U_K\), \(C_K\) be the class number, the unit group, and the cyclotomic units (in the sense of Sinnott) of \(K\). \textit{W. Sinnott} [Ann. Math. (2) 108, 107-134 (1978; Zbl 0395.12014)] showed that \([U_K : C_K]= 2^{n-1} c_K h_K\), where \(c_K\) is a certain rational number. One can write \(c_K=c_K' c_K''\), where NEWLINE\[NEWLINE c_K'=\frac{\prod_{\ell} [K\cap \mathbb Q(\zeta_{\ell^{\infty}}):\mathbb Q]} {[K : \mathbb Q]} NEWLINE\]NEWLINE and \(c_K''= [\mathbb Z[G] : Iw(K)]\). Here \(Iw(K)\) is a certain submodule of the group ring \(\mathbb Q[G]\). The present paper provides a character by character refinement of this result. Namely, let \(p\) be an odd prime and let \(\psi\) be a \(\mathbb Q_p\)-irreducible character of \(G\) of order prime to \(p\). Then NEWLINE\[NEWLINE [(U_K\otimes \mathbb Z_p)_{\psi} : (C_K\otimes \mathbb Z_p)_{\psi}] NEWLINE\]NEWLINE equals \(c_{K,\psi} h_{K,\psi}\) up to a \(p\)-adic unit, where \(c_{K,\psi}=c_{K,\psi}' c_{K,\psi}''\). Here NEWLINE\[NEWLINE c_{K,\psi}''= [\mathbb Z_p[G]_{\psi} : (Iw(K)\otimes \mathbb Z_p)_{\psi}]. NEWLINE\]NEWLINE Also, \(c_{K,\psi}'=1\) when \(\psi\neq 1\) and \(c_{K,\psi}'=c_K\) when \(\psi=1\). The proof of the analogous result for class groups using odd characters of imaginary Abelian fields uses the Main Conjecture, proved by \textit{B. Mazur} and \textit{A. Wiles} [Invent. Math. 76, 179-330 (1984; Zbl 0545.12005)]. The proof in the present paper also uses the Main Conjecture. A similar result has been given by \textit{L. Kuz'min} [Izv. Math. 60, No. 4, 695-761 (1996); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 60, No. 4, 43-110 (1996; Zbl 1007.11065)], with a different proof.