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

TL;DR

This paper constructs a natural morphism from inverse limits of units in $bk_n$ to a subset of linear functionals on $C^1(bz_p)$, with applications to Gauss sums and ideal class annihilation.

Contribution

It introduces a new $bz_p[[T-1]]$-morphism linking units in $bk_n$ to $C^1(bz_p)^*$, enabling novel interpolation and annihilation techniques.

Findings

Constructed a $bz_p[[T-1]]$-morphism from units to $C^1(bz_p)^*$

Applied to interpolate Gauss sums attached to Dirichlet characters

Enabled explicit annihilation of real ideal classes

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 $\varprojlim k_n^{\times} \otimes_{\mathbb{Z}} \mathbb{Z}_p$ into a special subset of $C^1(\mathbb{Z}_p)^*$, 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.