Let $\varphi :X\to X$ be a morphism of a variety defined over a number field $K$, let $V\subset X$ be a $K$-subvariety, and let ${𝒪}_{\varphi }\left(P\right)=\{{\varphi }^n\left(P\right):n\ge 0\}$ be the orbit of a point $P\in X\left(K\right)$. We describe a local-global principle for the intersection $V\cap {𝒪}_{\varphi }\left(P\right)$. This principle may be viewed as a dynamical analog of the Brauer–Manin obstruction. We show that the rational points of $V\left(K\right)$ are Brauer–Manin unobstructed for power maps on ${\mathbb{P}}^2$ in two cases: (1) $V$ is a translate of a torus. (2) $V$ is a line and $P$ has a preperiodic coordinate. A key tool in the proofs is the...

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...

Let K, L be algebraic number fields with K ⊆ L, and $O_K$, $O_L$ their respective rings of integers. We consider the trace map $T=T_{L/K}:L\to K$ and the $O_K$-ideal $T\left(O_L\right)\subseteq O_K$. By I(L/K) we denote the group indexof $T\left(O_L\right)$ in $O_K$ (i.e., the norm of $T\left(O_L\right)$ over ℚ). It seems to be difficult to determine I(L/K) in the general case. If K and L are absolutely abelian number fields, however, we obtain a fairly explicit description of the number I(L/K). This is a consequence of our description of the Galois module structure of $T\left(O_L\right)$ (Theorem 1)....

Let $p\ge 3$ be a prime. Let $n\in \mathbb{N}$ such that $n\ge 1$, let ${\chi }_1,...,{\chi }_n$ be characters of conductor $d$ not divided by $p$ and let $\omega$ be the Teichmüller character. For all $i$ between $1$ and $n$, for all $j$ between $0$ and $(p-3)/2$, set $${\theta }_{i,j}=\left\{\begin{array}{cc}{\chi }_i{\omega }^{2j+1}\hfill & \text{if}{\chi }_i\mathrm{is}\mathrm{odd};\hfill \\ {\chi }_i{\omega }^{2j}\hfill & \text{if}{\chi }_i\mathrm{is}\mathrm{even}.\hfill \end{array}\right.$$ Let $K={\mathbb{Q}}_p({\chi }_1,...,{\chi }_n)$ and let $\pi$ be a prime of the valuation ring ${𝒪}_K$ of $K$. For all $i,j$ let $f(T,{\theta }_{i,j})$ be the Iwasawa series associated to ${\theta }_{i,j}$ and $\overline{f(T,{\theta }_{i,j})}$ its reduction modulo $\left(\pi \right)$. Finally let $\overline{{𝔽}_p}$ be an algebraic closure of ${𝔽}_p$. Our main result is that if the characters ${\chi }_i$ are all distinct modulo $\left(\pi \right)$, then $1$ and the series...

Let $k$ be a finite field. Wintenberger used the field of norms to give an equivalence between a category whose objects are totally ramified abelian $p$-adic Lie extensions $E/F$, where $F$ is a local field with residue field $k$, and a category whose objects are pairs $(K,A)$, where $K\cong k\left(\right(T\left)\right)$ and $A$ is an abelian $p$-adic Lie subgroup of ${\mathrm{Aut}}_k\left(K\right)$. In this paper we extend this equivalence to allow $\mathrm{Gal}(E/F)$ and $A$ to be arbitrary abelian pro-$p$ groups.

Let $K$ be a field of degree $n$ over ${\mathbf{Q}}_p$, the field of rational $p$-adic numbers, say with residue degree $f$, ramification index $e$ and differential exponent $d$. Let $O$ be the ring of integers of $K$ and $\mathit{P}$ its unique prime ideal. The trace and norm maps for $K/{\mathbf{Q}}_p$ are denoted $Tr$ and $N$, respectively. Fix $q=p^r$, a power of a prime $p$, and let $\eta$ be a numerical character defined modulo $q$ and of order $o\left(\eta \right)$. The character $\eta$ extends to the ring of $p$-adic integers ${\mathbb{Z}}_p$ in the natural way; namely $\eta \left(u\right)=\eta \left(\tilde{u}\right)$, where $\tilde{u}$ denotes the residue...