Trace functions and Galois invariant \(p\)-adic measures (Q2458365)

In [J. Number Theory 88, No. 1, 13--38 (2001; Zbl 0965.11049)] \textit{V. Alexandru, N. Popescu} and \textit{A. Zaharescu} defined for an element \(\alpha\in\overline{{\mathbb Q}_ p}\), an algebraic closure of the \(p\)-adic number field \({\mathbb Q}_ p\), the absolute trace of \(\alpha\) over \({\mathbb Q}_ p\), \({\text{Tr}}(\alpha)= {{1}\over {\deg (\alpha)}} \text{tr} _ {{\mathbb Q}_ p (\alpha)/ {\mathbb Q}_ p}(\alpha)\) and they obtained that \(\text{Tr}(\alpha) =\int_{C(\alpha)} x\,d\pi_ \alpha\) where \(C(\alpha) = \{\sigma(\alpha)\mid \sigma\in G=\text{Gal}_{\text{cont}}({\mathbb C}_ p/{\mathbb Q} _ p)\}\) and \(\pi_ \alpha\) is a \(p\)-adic distribution. In this way, it was defined for \(T\in{\mathbb C}_ p\), \(\text{Tr} (T)=\int_ {C(T)}x \,d\pi_ T\) if the integral exists. For \(z\not\in C(1/T)\), \(T\in {\mathbb C}_ p\) such that the integral exists, it was defined \(F(T,z)=\int_{C(T)} {{1}\over{1-xz}}\,d\pi_ T(x)\). The main goal of the paper under review is to study some generalizations of the above objects. For a subset \(M\) of \({\mathbb C}_ p\) that is \(G\)-invariant and compact and \(\mu\) a \(G\)-invariant probability distribution on \(M\), the authors define the trace of \(\mu\) by \(\text{Tr}\mu =\int_ M x\,d\mu(x)\) if the integral exists. They associate a trace function \(F(\mu,z)=\int _ M {{1}\over{1-zx}}\,d\mu(x)\) for all \(z\in{\mathbb C}_ p\) such that the integral exists. This analytic object provides significant algebraic information about \({\mathbb C}_ p\). For instance, if \(E\) is a closed subfield of \({\mathbb C}_ p\) on which the trace map \(\text{Tr}\) is defined and continuous, and if \(E={\mathbb Q}_ p (T)\), then the trace on the entire field \(E\) is determined by the Taylor series expansion \(F(T,z) = \sum _ {n=0}^{\infty} \operatorname{Tr}T^ n z^ n\). If \(M\) and \(\mu\) are \(G\)-invariant, the authors use the action of the Galois group to express and compute the above integrals.