A representation theorem for a class of rigid analytic functions
Journal de théorie des nombres de Bordeaux, Volume 15 (2003) no. 3, p. 639-650

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

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

@article{JTNB_2003__15_3_639_0,
     author = {Alexandru, Victor and Popescu, Nicolae and Zaharescu, Alexandru},
     title = {A representation theorem for a class of rigid analytic functions},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {15},
     number = {3},
     year = {2003},
     pages = {639-650},
     zbl = {1070.11053},
     mrnumber = {2142227},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2003__15_3_639_0}
}
Alexandru, Victor; Popescu, Nicolae; Zaharescu, Alexandru. A representation theorem for a class of rigid analytic functions. Journal de théorie des nombres de Bordeaux, Volume 15 (2003) no. 3, pp. 639-650. http://www.numdam.org/item/JTNB_2003__15_3_639_0/

[APZ1] V. Alexandru, N. Popescu, A. Zaharescu, On closed subfields of Cp. J. Number Theory 68 (1998), 131-150. | MR 1605907 | Zbl 0901.11035

[APZ2] V. Alexandru, N. Popescu, A. Zaharescu, Trace on Cp. J. Number Theory 88 (2001), 13-48. | MR 1825989 | Zbl 0965.11049

[Am] Y. Amice, Les nombres p-adiques, Presse Univ. de France, Collection Sup. 1975. | MR 447195 | Zbl 0313.12104

[Ax] J. Ax, Zeros of Polynomials over Local Fields - The Galois Action, J. Algebra 15 (1970), 417-428. | MR 263786 | Zbl 0216.04703

[PZ] N. Popescu, A. Zaharescu, On the main invariant of an element over a local field, Portugaliae Mathematica 54 (1997), 73-83. | MR 1440129 | Zbl 0894.11045

[Ar] E. Artin, Algebraic Numbers and Algebraic Functions, Gordon and Breach, New York/London/ Paris, 1967. | MR 237460 | Zbl 0194.35301

[FP] J. Fresnel, M. Van Der Put, Géometrie Analytique Rigide et Applications, Birkhäuser. Boston. Basel. Stuttgart, 1981. | MR 644799 | Zbl 0479.14015