### A class number criterion for the equation $({x}^{p}-1)/(x-1)=p{y}^{q}$

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Introduction. Recently J. Rutkowski (see [3]) has defined the p-adic analogue of the Walsh system, which we shall denote by ${\left(\varphi \u2098\right)}_{m\in \mathbb{N}\u2080}$. The system ${\left(\varphi \u2098\right)}_{m\in \mathbb{N}\u2080}$ is defined in the space C(ℤₚ,ℂₚ) of ℂₚ-valued continuous functions on ℤₚ. J. Rutkowski has also considered some questions concerning expansions of functions from C(ℤₚ,ℂₚ) with respect to ${\left(\varphi \u2098\right)}_{m\in \mathbb{N}\u2080}$. This paper is a remark to Rutkowski’s paper. We define another system ${\left(h\u2099\right)}_{n\in \mathbb{N}\u2080}$ in C(ℤₚ,ℂₚ), investigate its properties and compare it to the system defined by Rutkowski. The system...

Let $p$ be a prime number, ${\mathbb{Q}}_{p}$ the field of $p$-adic numbers and ${\u2102}_{p}$ the completion of the algebraic closure of ${\mathbb{Q}}_{p}$. In this paper we obtain a representation theorem for rigid analytic functions on ${\mathbf{P}}^{1}\left({\u2102}_{p}\right)\setminus C(t,\u03f5)$ which are equivariant with respect to the Galois group $G=Ga{l}_{cont}({\u2102}_{p}/{\mathbb{Q}}_{p})$, where $t$ is a lipschitzian element of ${\u2102}_{p}$ and $C(t,\u03f5)$ denotes the $\u03f5$-neighborhood of the $G$-orbit of $t$.