In the $p^n$-th cyclotomic field ${\mathbb{Q}}_{p^n},p$ a prime number, $n\in \mathbb{N}$, the prime $p$ is totally ramified and the only ideal above $p$ is generated by ${\omega }_n={\zeta }_{p^n}-1$, with the primitive $p^n$-th root of unity ${\zeta }_{p^n}=e^{\frac{2\pi i}{p^n}}$. Moreover these numbers represent a norm coherent set, i.e. ${\mathbf{\text{N}}}_{{\mathbb{Q}}_{p^{n+1}}}/{\mathbb{Q}}_{p^n}\left({\omega }_{n+1}\right)={\omega }_n$. It is the aim of this article to establish a similar result for the ray class field $K_{{𝔭}^n}$ of conductor ${𝔭}^n$ over an imaginary quadratic number field $K$ where ${𝔭}^n$ is the power of a prime ideal in $K$. Therefore the exponential function has to be replaced by a suitable elliptic...

Let $X$ be the quotient group of the $S$-adele ring of an algebraic number field by the discrete group of $S$-integers. Given a probability measure $\mu$ on $X^d$ and an endomorphism $T$ of $X^d$, we consider the relation between uniform distribution of the sequence $T^n𝐱$ for $\mu$-almost all $𝐱\in X^d$ and the behavior of $\mu$ relative to the translations by some rational subgroups of $X^d$. The main result of this note is an extension of the corresponding result for the $d$-dimensional torus ${𝕋}^d$ due to B. Host.

Values of $\lambda$ are determined for which there exist positive solutions of the system of three-point boundary value problems, $u^{\text{'}\text{'}}+\lambda a\left(t\right)f\left(v\right)=0$, $v^{\text{'}\text{'}}+\lambda b\left(t\right)g\left(u\right)=0$, for $0<t<1$, and satisfying, $u\left(0\right)=\beta u\left(\eta \right)$, $u\left(1\right)=\alpha u\left(\eta \right)$, $v\left(0\right)=\beta v\left(\eta \right)$, $v\left(1\right)=\alpha v\left(\eta \right)$. A Guo-Krasnosel’skii fixed point theorem is applied.