A representation theorem for a class of rigid analytic functions

Victor Alexandru; Nicolae Popescu; Alexandru Zaharescu

Victor Alexandru ; Nicolae Popescu ; Alexandru Zaharescu

Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 3, pp. 639-650.

Résumé
Abstract

Soit $p$ un nombre premier, $ℚ_{p}$ le corps des nombres $p$ -adiques et $ℂ_{p}$ la complétion d’une clôture algébrique de $ℚ_{p}$ . Dans cet article, nous obtenons un théorème de représentation pour les fonctions analytiques rigides sur $𝐏^{1} (ℂ_{p}) ∖ C (t, ϵ)$ qui sont équivariantes par le groupe de Galois $G = G a l_{c o n t} (ℂ_{p} / ℚ_{p})$ , où $t$ désigne un élément lipschitzien de $ℂ_{p}$ et $C (t, ϵ)$ un $ϵ$ -voisinage de la $G$ -orbite de $t$ .

Let $p$ be a prime number, $ℚ_{p}$ the field of $p$ -adic numbers and $ℂ_{p}$ the completion of the algebraic closure of $ℚ_{p}$ . In this paper we obtain a representation theorem for rigid analytic functions on $𝐏^{1} (ℂ_{p}) ∖ C (t, ϵ)$ which are equivariant with respect to the Galois group $G = G a l_{c o n t} (ℂ_{p} / ℚ_{p})$ , where $t$ is a lipschitzian element of $ℂ_{p}$ and $C (t, ϵ)$ denotes the $ϵ$ -neighborhood of the $G$ -orbit of $t$ .

MR Zbl

@article{JTNB_2003__15_3_639_0,
     author = {Victor Alexandru and Nicolae Popescu and Alexandru Zaharescu},
     title = {A representation theorem for a class of rigid analytic functions},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {639--650},
     publisher = {Universit\'e Bordeaux I},
     volume = {15},
     number = {3},
     year = {2003},
     zbl = {1070.11053},
     mrnumber = {2142227},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2003__15_3_639_0/}
}

TY  - JOUR
AU  - Victor Alexandru
AU  - Nicolae Popescu
AU  - Alexandru Zaharescu
TI  - A representation theorem for a class of rigid analytic functions
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2003
SP  - 639
EP  - 650
VL  - 15
IS  - 3
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/item/JTNB_2003__15_3_639_0/
LA  - en
ID  - JTNB_2003__15_3_639_0
ER  -

%0 Journal Article
%A Victor Alexandru
%A Nicolae Popescu
%A Alexandru Zaharescu
%T A representation theorem for a class of rigid analytic functions
%J Journal de théorie des nombres de Bordeaux
%D 2003
%P 639-650
%V 15
%N 3
%I Université Bordeaux I
%U https://jtnb.centre-mersenne.org/item/JTNB_2003__15_3_639_0/
%G en
%F JTNB_2003__15_3_639_0

Victor Alexandru; Nicolae Popescu; Alexandru Zaharescu. A representation theorem for a class of rigid analytic functions. Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 3, pp. 639-650. https://jtnb.centre-mersenne.org/item/JTNB_2003__15_3_639_0/

Bibliographie
Cité par

[APZ1] V. Alexandru, N. Popescu, A. Zaharescu, On closed subfields of Cp. J. Number Theory 68 (1998), 131-150. | MR | Zbl

[APZ2] V. Alexandru, N. Popescu, A. Zaharescu, Trace on Cp. J. Number Theory 88 (2001), 13-48. | MR | Zbl

[Am] Y. Amice, Les nombres p-adiques, Presse Univ. de France, Collection Sup. 1975. | MR | Zbl

[Ax] J. Ax, Zeros of Polynomials over Local Fields - The Galois Action, J. Algebra 15 (1970), 417-428. | MR | Zbl

[PZ] N. Popescu, A. Zaharescu, On the main invariant of an element over a local field, Portugaliae Mathematica 54 (1997), 73-83. | MR | Zbl

[Ar] E. Artin, Algebraic Numbers and Algebraic Functions, Gordon and Breach, New York/London/ Paris, 1967. | MR | Zbl

[FP] J. Fresnel, M. Van Der Put, Géometrie Analytique Rigide et Applications, Birkhäuser. Boston. Basel. Stuttgart, 1981. | MR | Zbl