On some congruences for units of local \(p\)-cyclotomic fields (Q1349504)

Let \(p\) be prime, \(k_n\) the local \(p^{n+1}\)th cyclotomic field, \(K_\infty\) the union of these \(k_n\), \(U_n\) the group of principal units of \(k_n\), and \(B_n\) its subgroup of all units which are norms from every \(U_m\), \(m\geq n\). Moreover, let \(\chi\) be a \(\mathbb{Q}_p\)-valued character of \(\Delta=\text{Gal}(k_0/\mathbb{Q}_p)\), \(\Gamma=\text{Gal}(k_\infty/k_0)\) and \(\Lambda=\mathbb{Z}_p[[\Gamma]]\). The author studies the submodule \(J(n,a,\chi)\) of the \(\chi\)-eigenspace \(B_n(\chi)\) consisting of all units which are congruent to 1 modulo \((1-\xi)^a\), where \(\xi\) is a primitive \(p\)th root of unity. This generalizes earlier results of his [J. Reine Angew. Math. 462, 169-184 (1995; Zbl 0815.11055)], where the special case \(a=1\) was studied.