Cohomology groups of circular units (Q2731058)

Let \(k\) be a real abelian field of conductor \(f\) and \(\cup_{n\geq 0}k_n\) its \(\mathbb Z_p\)-extension, where \(p\) is an odd prime not dividing \(f\varphi(f)\). Put \(G_{m,n}=\text{Gal}(k_m/k_n)\) for \(m>n\geq 0\) and let \(C_n\) be the group of circular units of \(k_n\) as defined by \textit{W. Sinnott} [Invent. Math. 62, 181-234 (1980; Zbl 0465.12001)]. The following theorem is proved:NEWLINENEWLINENEWLINE(1) \(C_m^{G_{m,n}}=C_n\);NEWLINENEWLINENEWLINE(2) \({\widehat H}^i(G_{m,n},C_m)\simeq(\mathbb Z/p^{m-n}\mathbb Z)^{\ell-\delta}\), where \(\ell\) is the number of prime ideals of \(k\) above \(p\) and \(\delta\) is 0 or 1 according as \(i\) is odd or even.NEWLINENEWLINENEWLINESpecial cases where the conductor \(f\) is prime or divisible by two primes were studied by the first author [Tôhoku Math. J. (2) 51, 305-313 (1999; Zbl 0948.11039)] and \textit{J.-R. Belliard} [J. Number Theory 69, 16-49 (1998; Zbl 0911.11051)]. The above theorem generalizes previous results and shows that the question for circular units is simpler than for the full unit group, investigated by \textit{K. Iwasawa} [Am. J. Math. 105, 189-200 (1983; Zbl 0525.12009)]. The basic ideas are taken from a paper of the first author [J. Algebra 152, 514-519 (1992; Zbl 0776.11066)].