The \(p\)-adic measure on the orbit of an element of \(\mathbb{C}_p\)
type: Item
instance of: scholarly article
title: The \(p\)-adic measure on the orbit of an element of \(\mathbb{C}_p\)
zbMATH Open document ID: 1164.11072
author: V. Alexandru
author: N. Popescu
author: M. Vâjâitu
author: Alexandru Zaharescu
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2008-12-01T00:00:00Z
full work available at URL: https://eudml.org/doc/108722
review text: Let \(p\) be a prime number, \({\mathbb Q}_p\) the field of \(p\)-adic numbers, and \({\mathbb C}_p\) the completion of some algebraic closure of \({\mathbb Q}_p\). For each \(x \in {\mathbb C}_p\), let \(O(x)\) denote its orbit with respect to the Galois group \(G = \text{Gal}({\mathbb C}_p/{\mathbb Q}_p)\), the continuous \(\mathbb{Q}_p\)-automorphisms of \(\mathbb{C}_p\). Finally, let \(E(x) = ({\mathbb C}_p \cup \{\infty\}) \setminus O(x)\) denote the complement of the orbit \(O(x)\). The authors are interested in the behaviour of rigid analytic functions on \(E(x)\), and to this end study functionals on the \(\mathbb{C}_p\)-algebra \({\mathcal C}(O(x),\mathbb{C}_p)\) of continuous functions from \(O(x)\) to \(\mathbb{C}_p\). After some preliminary sections, they prove the following result: Every \(\mathbb{Q}_p\)-functional \(\phi: {\mathcal C}_G(O(x),{\mathbb C}_p) \rightarrow \mathbb{Q}_p\), where \({\mathcal C}_G(O(x),{\mathbb C}_p)\) is the subset of \(G\)-equivariant functionals of \({\mathcal C}(O(x),{\mathbb C}_p)\), can be extended uniquely to a \(\mathbb{C}_p\)-functional \(\widetilde{\phi}: {\mathcal C}(O(x),\mathbb{C}_p) \rightarrow \mathbb{C}_p\), and this functional gives rise to an equivariant measure \(\mu_\phi\) on \(O(x)\) with values in \({\mathbb Q}_p\), the algebraic closure of \({\mathbb Q}_p\) inside \({\mathbb C}_p\). The next few sections are dedicated to Cauchy transforms of functions with respect to a measure, and in the final section the authors present an analogue of a theorem of \textit{D. Barsky} [Math. Ann. 232, 255--266 (1978; Zbl 0352.12014)], which relates measures on \(O(x)\) with a suitable class of rigid analytic functions on the complement of \(O(x)\).
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11S99
zbMATH DE Number: 5376053
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5376053