Semi-local units at p of a cyclotomic ${\mathbb Z}_p$-extension congruent to 1 modulo $\zeta_p - 1$

Abstract

Let p be a prime number. Let K be an abelian number field with p ∤ [K : ℚ] and ζ_p ∈ K, K_∞/K the cyclotomic ℤ_p-extension, and K_n the nth layer with K₀ = K. Let $\mathcal U$_n be the group of semi-local principal units of K_n at the prime p, and $\mathcal U$_n⁽¹⁾ the elements u of $\mathcal U$_n satisfying the congruence u ≣ 1 modulo ζ_p - 1. The Galois module structure of $\mathcal U$_n is well understood. The purpose of this paper is to determine the Galois module structure of $\mathcal U$_n⁽¹⁾.