Representation results for equivariant rigid analytic functions
Let \({\mathbb C}_p\) be the completion of an algebraic closure \(\bar{\mathbb Q}_p\) of the field of \(p\)-adic numbers. Let \(G:=\mathrm{Gal}(\bar{\mathbb Q}_p/ {\mathbb Q})\) which is isomorphic with the group \(\mathrm{Gal}_{\mathrm{cont}}({\mathbb C}_p/{\mathbb Q}_p)\) of continuous automorphisms of \({\mathbb C}_p\) over \({\mathbb Q}_p\). Let \({\mathcal O}(x)\) be the orbit of an element of \(x\in {\mathbb C}_p\) with respect \(G\). A rigid analytic function \(f\) defined on \(D \subseteq {\mathbb C}_p\) is \textit{\(G\)-equivariant} if \(f(\sigma(z))=\sigma(f(z))\) for all \(z\in D\) and all \(\sigma \in G\). The objective of this paper is to find representation results for equivariant rigid analytic functions defined on \(E(x):={\mathbb C}_p \cup \{\infty\}\setminus {\mathcal O}(x)\)with values in \({\mathbb C}_p\) that vanish at \(\infty\). Let \(f\) be a \(p\)-adic \(G\)-equivariant rigid analytic function on \(E(x)\) which vanishes at \(\infty\). The paper is divided in two parts. The first one studies the case \(x\) algebraic over \({\mathbb Q}_p\), that is \(x\in \bar{\mathbb Q}_p\). Then there exists a sequence \(\mu:=\{\mu_n\}_{n\geq 1}\) of equivariant measures on \({\mathcal O}(x)\) such that \(\lim_{n\to\infty}\|\mu_n\|^{1/n}=0\) and \(f=f_{\mu}:= \sum_{n=1}^{\infty}\int_{{\mathcal O}(x)}\frac{1}{(z-t)^n} d\mu_n(t)\). The second part considers \(x\in {\mathbb C}_p\) transcendent. In this case \(f\) is uniformly approximated in a specific norm by a sequence from a \({\mathbb C}_p\)--vector space generated by some special integral functions.
scientific article; zbMATH DE number 6053982