Let $B$ be a quaternion algebra over a number field $k$. To a pair of Hilbert symbols $\{a,b\}$ and $\{c,d\}$ for $B$ we associate an invariant $\rho ={\rho }_R\left(\left[𝒟\right(a,b\left)\right],\left[𝒟\right(c,d\left)\right]\right)$ in a quotient of the narrow ideal class group of $k$. This invariant arises from the study of finite subgroups of maximal arithmetic kleinian groups. It measures the distance between orders $𝒟(a,b)$ and $𝒟(c,d)$ in $B$ associated to $\{a,b\}$ and $\{c,d\}.$ If $a=c$, we compute ${\rho }_R(\left[𝒟(a,b)\right],\left[𝒟(c,d)\right])$ by means of arithmetic in the field $k\left(\sqrt{a}\right).$ The problem of extending this algorithm to the general case leads to studying a finite graph associated...

Let $p$ be a prime number. We say that a number field $F$ satisfies the condition $\left(H_{p^n}^{\prime }\right)$ when any abelian extension $N/F$ of exponent dividing $p^n$ has a normal integral basis with respect to the ring of $p$-integers. We also say that $F$ satisfies $\left(H_{p^{\infty }}^{\prime }\right)$ when it satisfies $\left(H_{p^n}^{\prime }\right)$ for all $n\ge 1$. It is known that the rationals $\mathbb{Q}$ satisfy $\left(H_{p^{\infty }}^{\prime }\right)$ for all prime numbers $p$. In this paper, we give a simple condition for a number field $F$ to satisfy $\left(H_{p^n}^{\prime }\right)$ in terms of the ideal class group of $K=F\left({\zeta }_{p^n}\right)$ and a “Stickelberger ideal” associated to the Galois group...

Let $K$ be a $p$-adic local field with residue field $k$ such that $[k:k^p]=p^e\<+\infty$ and $V$ be a $p$-adic representation of $\text{Gal}(\overline{K}/K)$. Then, by using the theory of $p$-adic differential modules, we show that $V$ is a Hodge-Tate (resp. de Rham) representation of $\text{Gal}(\overline{K}/K)$ if and only if $V$ is a Hodge-Tate (resp. de Rham) representation of $\text{Gal}(\overline{K^{\text{pf}}}/K^{\text{pf}})$ where $K^{\text{pf}}/K$ is a certain $p$-adic local field with residue field the smallest perfect field $k^{\text{pf}}$ containing $k$.