On a construction of $C^1({\mathbb{Z}}_p)$ functionals from ${\mathbb{Z}}_p$-extensions of algebraic number fields

Abstract

Let $k$ be any number field and $k_{\infty }/k$ any ${\mathbb{Z}}_p$-extension. We construct a natural $\Lambda ={\mathbb{Z}}_p[[T-1]]$-morphism from $\underleftarrow{\lim }{k}_n^{\times }{\otimes }_{\mathbb{Z}}{\mathbb{Z}}_p$ into a special subset of $C^1({\mathbb{Z}}_p{)}^{\ast }$, the collection of linear functionals on the set of continuously differentiable functions from ${\mathbb{Z}}_p\to {\mathbb{C}}_p$. We apply the results to the problem of interpolating Gauss sums attached to Dirichlet characters and the explicit annihilation of real ideal classes.