Good elements and metric invariants in \(B_{dR}^+\)
For any prime number \(p\), let \({\mathbb Q}_ p\) denote the field of \(p\)-adic numbers, \(\overline{ {\mathbb Q}_ p}\) an algebraic closure of \({\mathbb Q} _ p\) and \({\mathbb C}_ p\) the \(p\)-adic field of complex numbers. Let \(B ^ + _ {dR}\) be the ring of \(p\)-adic periods of algebraic varieties defined over \(p\)-adic fields as considered by \textit{J.-M. Fontaine} in [Ann. Math. (2) 115, 529--577 (1982; Zbl 0544.14016)], \(B ^ + _ {dR}\) is a topological local ring with maximal ideal \(I\) and residue field \({\mathbb C}_ p\) and it is endowed with a canonical, continuous action of \(G:=\text{ Gal}({\overline{{\mathbb Q}_ p}}/{\mathbb Q} _ p)\). Then \(B ^ + _ {dR}\) and \(B _ n := B ^ + _ {dR}/I ^ n\), \(n\geq 1\), are canonically \({\overline{{\mathbb Q}_ p}}\)-algebras with \({\overline{{\mathbb Q}_ p}}\) dense in \(B ^ + _ {dR}\) and \(B _ n\) in the natural topology, which is finer than the valuation ring topology. For each \(n\), let \(w_ n\) be the canonical valuation defined on \({\overline{{\mathbb Q}_ p}}\) as considered by \textit{A. Iovita} and \textit{A. Zaharescu} in [Compos. Math. 117, No. 1, 1--31 (1999; Zbl 0943.11054)], \(w _ n\) induces the canonical topology in \(B_ n\). An element \(\alpha \in {\overline{{\mathbb Q}_ p^ \ast}}\) is called ``good'' at level \(n\) if \(w_ n(\alpha)=v(\alpha)\), where \(v\) is the canonical \(p\)-adic valuation on \({\overline{{\mathbb Q}_ p}}\). Let \({\mathcal G} _ n\) be the subgroup of good elements at level \(n\). Then \({\mathcal G}_ n \supseteq {\mathcal G}_ {n+1}\) for all \(n\). Define \({\mathcal H}_ n := {\overline{{\mathbb Q}_ p ^ \ast}}/ {\mathcal G}_ n\). Then \({\mathcal H} _ 1 \cong {\mathbb Q}/{\mathbb Z}\). The main results in this paper are: (1) for any \(n\geq 1\), \({\mathcal H}_ n\) is a torsion group; (2) \({\mathcal G}_ 1/{\mathcal G}_ n\) is a \(p\)-group; (3) for any \(Z\in B_ {dR}^ +\) whose projection in \({\mathbb C}_ p\) is transcendental over \({\mathbb Q}_ p\), the sequence \(\{\ell_ n(Z)\}\) is constant, where \(\ell _ n(Z) \in {\mathbb R} \cup \{\infty\}\) is a certain metric invariant of \(Z\).
scientific article; zbMATH DE number 2124031