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Zbl 1070.11053
Alexandru, Victor; Popescu, Nicolae; Zaharescu, Alexandru
A representation theorem for a class of rigid analytic functions. (English)
[J] J. Théor. Nombres Bordx. 15, No. 3, 639-650 (2003). ISSN 1246-7405

For a prime number $p$, let ${\Bbb Q}\sb p$ denote the field of $p$-adic numbers and ${\Bbb C} \sb p$ the completion of a fixed algebraic closure $\overline{\Bbb Q}\sb p$ of ${\Bbb Q}\sb p$. Let $G$ be the absolute Galois group of ${\Bbb Q}\sb p$: $G=\text{Gal}(\overline{\Bbb Q}\sb p/{\Bbb Q}\sb p)$. The group $G$ is canonically isomorphic with the group of all continuous automorphisms of ${\Bbb C} \sb p$ over ${\Bbb Q}\sb p$. The purpose of the paper under review is to obtain a representation theorem for rigid analytic functions on $E(t,\varepsilon)= ({\Bbb C}\sb p \cup \infty)\setminus C(t,\varepsilon)$ which are equivariant with respect $G$, where $t$ is a {\sl{Lipschitzian}} element of ${\Bbb C}\sb p$ and $C(t,\varepsilon)$ denotes the $\varepsilon$-neighborhood of the orbit of $t$ under the action of $G$. When $t$ is algebraic over ${\Bbb Q}\sb p$, these functions can be described easily: sending $t$ to the point at infinity, they correspond to the power series $\sum\sb{n=0}\sp{\infty} a\sb n z\sp n$ with $a\sb n\in {\Bbb Q}\sb p$ and $\lim\sb{n\to\infty} \root n \of {\vert a\sb n\vert }=0$. It $t$ is transcendental over ${\Bbb Q}\sb p$ it is not obvious that there exist nonconstant equivariant rigid analytic functions on $E(t,\varepsilon)$. However, for Lipschitzian elements, the authors [J. Number Theory 88, No. 1, 13--38 (2001; Zbl 0965.11049)] construct such a function $z\mapsto F(t,z)$. In this paper rigid analytic functions $F\sb {m,n} (t,z)$ are defined on $E(t,\varepsilon)$ for any Lipschitzian element $t$ and any nonnegative integer numbers $m,n$. Then any equivariant rigid analytic function on $E(t,\varepsilon)$ is expressed in terms of these functions $F\sb{m,n}(t,z)$.
[Gabriel D. Villa-Salvador (México D.F.)]

MSC 2000:: *11S80 Other analytic theory of local fields
12F20 Transcendental extensions
12J10 Valued fields
14G22 Rigid analytic geometry

Keywords: rigid analytic functions; transcendental elements; Lipschitzian elements; $p$-adic fields

Citations: Zbl 0965.11049

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