All, Waller: On a construction of $C^1(\mathbb{Z}_p)$ functionals from $\mathbb{Z}

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Timothy All; Bradley Waller
On a construction of $C^1(\mathbb{Z}_p)$ functionals from $\mathbb{Z}_p$-extensions of algebraic number fields
Journal de théorie des nombres de Bordeaux, 29 no. 1 (2017), p. 29-50, doi: 10.5802/jtnb.968
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Class. Math.: 11R23
Keywords: distributions, $L$-functions, Gauss sums, class group

Résumé - Abstract

Let $k$ be any number field, and let $k_{\infty }/k$ be any $\mathbb{Z}_p$-extension. We construct a natural $\mathbb{Z}_p\llbracket T-1 \rrbracket $-morphism from $\varprojlim k_n^{\times } \otimes _{\mathbb{Z}} \mathbb{Z}_p$ into a special subset of $C^1(\mathbb{Z}_p)^*$, the dual of the $\mathbb{C}_p$-vector space of continuously differentiable functions from $\mathbb{Z}_p \rightarrow \mathbb{C}_p$. We apply the results to the problem of interpolating Gauss sums attached to Dirichlet characters.

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