The \(p\)-adic measure on the orbit of an element of \(\mathbb{C}_p\)
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 \(O(x)\).
scientific article; zbMATH DE number 5376053