Cyclotomic units in \(\mathbb Z_p\)-extensions
Let \(k\) be an abelian field and let \(\zeta_m\) denote a primitive \(m\)-th root of 1. In the unit group \(E_k\) of \(k\), the units generated by \(-1\) and the norms of all numbers \(1- \zeta_m^a\) (from any possible field \(\mathbb Q (\zeta_m)\)) constitute a subgroup, the so-called circular units in Sinnott's sense, denoted by \(C_k\). The authors call another subgroup of \(E_k\) the Washington group of cyclotomic units and denote it by \(\overline {C}_k\); this is defined as \(C_{\mathbb Q (\zeta_m)} \cap E_k\), where \(m\) is the conductor of \(k\). The group \(C_k\) is contained in \(\overline {C}_k\). For a prime \(p\), let \(K_n\) denote the \(n\)-th layer \((n\geq 0)\) in the cyclotomic \(\mathbb Z_ p\)-extension \(K/k\). Denote the projective limits (with respect to norms) of \(C_{K_n} \otimes \mathbb Z_ p\) and \(\overline {C}_{K_n} \otimes \mathbb Z_p\) by \(C\) and \(\overline {C}\), respectively. The authors prove that the index \([\overline {C}: C]\) is finite. In fact, they prove the following theorem: For \(n\) sufficiently large, \([ E_{K_n}: \overline {C}_{K_n}]= cp^{t_n} h^+_{K_n}\) with \(0\leq t_ n< d\), where \(c\) and \(d\) are rational numbers depending only on \(K\). Here \(h^+_{K_n}\) denotes the class number of the maximal real subfield of \(K_n\). The finiteness result follows on combining this theorem with \textit{W. Sinnott}'s results about \(C_{K_n}\) [Invent. Math. 62, 181--234 (1980; Zbl 0465.12001)]. The proof of the theorem makes use of sophisticated techniques developed by Sinnott [loc. cit.].
scientific article; zbMATH DE number 735311