On some congruences for units of localp-cyclotomic fields

Abstract

Let p be a fixed odd prime number, and let kn (n > 0) be the local pn+l-st cyclotomic field Qp(#p,+~) and k~ = t2 kn. Let lIn be the group of principal units of kn and let ~B, be the intersection of Nm/nlIm of all m with m >_ n, Nm/n being the norm map from km x to kn x. Further, let 1I = [~lI, = ~_~Bn be the projective limit w.r.t, the relative norms. The Galois groups A = Gal(ko/lI~p) and F = Gal(k~/ko) act on these groups in a natural way. We fix the topological generator 7 of F such that ~ = ~l+p for all pa-th roots r of unity (a >_ 1). We identify, as usual, the completed group ring 7Zv[[F]] with the power series ring A = Zp[[S]] by ~ = 1 + s. For a (Qp-valued) character X of A, denote by U(X), Iln(X) and ~Bn(X) the x-eigenspaces of 1[, lI. and ~Bn respectively. They are considered as modules over A in a natural way. In [8], the A-module structures of U(X) and ~Bn(X) were determined (see the Fact in Section 2). The purpose of the present article is to determine the A-module structure of the submodule