Let $h_n$ denote the class number of $n$-th layer of the cyclotomic ${\mathbb{Z}}_2$-extension of $\mathbb{Q}$. Weber proved that $h_n(n\ge 1)$ is odd and Horie proved that $h_n(n\ge 1)$ is not divisible by a prime number $\ell$ satisfying $\ell \equiv 3,5(mod8)$. In a previous paper, the authors showed that $h_n(n\ge 1)$ is not divisible by a prime number $\ell$ less than ${10}^7$. In this paper, by investigating properties of a special unit more precisely, we show that $h_n(n\ge 1)$ is not divisible by a prime number $\ell$ less than $1.2·{10}^8$. Our argument also leads to the conclusion that $h_n(n\ge 1)$ is not divisible by...

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 $k$ be the algebraic closure of ${𝔽}_p$ and $K$ be the local field of formal power series with coefficients in $k$. The aim of this paper is the description of the set ${𝒴}_n$ of conjugacy classes of series of order $p^n$ for the composition law. This work is concerned with the formal power series with coefficients in a field of characteristic $p$ which are invertible and of finite order $p^n$ for the composition law. In order to investigate Oort’s conjecture, I give a description of conjugacy classes of series...

For a prime $p$ and positive integers $\ell \<k\<h\<p$ with $d=(h,k,\ell ,p-1)$, we show that $M$, the number of simultaneous solutions $x,y,z,w$ in ${\mathbb{Z}}_p^{*}$ to $x^h+y^h=z^h+w^h$, $x^k+y^k=z^k+w^k$, $x^{\ell }+y^{\ell }=z^{\ell }+w^{\ell }$, satisfies $${\displaystyle M\le 3d^2{(p-1)}^2+25hk\ell (p-1).}$$ When $hk\ell =o\left(p{d}^2\right)$ we obtain a precise asymptotic count on $M$. This leads to the new twisted exponential sum bound $${\displaystyle \left|\sum _{x=1}^{p-1}\chi \left(x\right)e^{2\pi if\left(x\right)/p}\right|\le 3^{\frac{1}{4}}{d}^{\frac{1}{2}}{p}^{\frac{7}{8}}+\sqrt{5}{\left(hk\ell \right)}^{\frac{1}{4}}{p}^{\frac{5}{8}},}$$ for trinomials $f=a{x}^h+b{x}^k+c{x}^{\ell }$, and to results on the average size of such sums.

Let $X$ and $Y$ be vector spaces over the same field $K$. Following the terminology of Richard Arens [Pacific J. Math. 11 (1961), 9–23], a relation $F$ of $X$ into $Y$ is called linear if $\lambda F\left(x\right)\subset F\left(\lambda x\right)$ and $F\left(x\right)+F\left(y\right)\subset F(x+y)$ for all $\lambda \in K\setminus \left\{0\right\}$ and $x,y\in X$. After improving and supplementing some former results on linear relations, we show that a relation $\Phi$ of a linearly independent subset $E$ of $X$ into $Y$ can be extended to a linear relation $F$ of $X$ into $Y$ if and only if there exists a linear subspace $Z$ of $Y$ such that $\Phi \left(e\right)\in Y|Z$ for all $e\in E$. Moreover, if...