Good elements and metric invariants in $B^+ _{dR}$

Abstract

Let $p$ be a prime, $\mathbb{Q}_{p}$ the field of $p$-adic numbers and $\mathbb{\bar{Q}}_{p}$ a fixed algebraic closure of $\mathbb{Q}_{p}$. $B_{dR}^{+}$ is the ring of $p$-adic periods of algebraic varieties over $p$-adic fields introduced by Fontaine. For each n one defines a canonical valuation $w_{n}$ on $\mathbb{\bar{Q}}_{p}$ such that $B_{dR}^{+}/I^{n}$ becomes the completion of $\mathbb{\bar{Q}}_{p}$ with respect to $w_{n}$, where $I$ is the maximal ideal of $B_{dR}^{+}$. An element $\alpha \in \mathbb{\bar{Q}}_{p}^{*}$ is said to be good at level $n$ if $w_{n}(\alpha ) = v(\alpha )$ where $v$ denotes the $p$-adic valuation on $\mathbb{\bar{Q}}_{p}$. The set $\mathcal{G}_{n}$ of good elements at level n is a subgroup of $\mathbb{\bar{Q}}_{p}^{*}$. We prove that each quotient group $\mathbb{\bar{Q}}_{p}^{*}/\mathcal{G}_{n}$ is a torsion group and that each quotient $\mathcal{G}_{1}/\mathcal{G}_{n}$ is a $p$-group. We also show that a certain sequence of metric invariants $\{ l_{n}(Z)\}_{n\in \mathbb{N}}$ associated to an element $Z \in B_{dR}^{+}$, is constant.