### Abstract

Let p be a prime, ℚ_{p} the field of p-adic numbers and ℚ̄_{p} a fixed algebraic closure of 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 ℚ̄_{p} such that B _{dR} ^{+}n/I^{n} becomes the completion of ℚ̄_{p} with respect to w_{n}, where I is the maximal ideal of B_{dR} ^{+}. An element α ∈ ℚ̄ _{p}* is said to be good at level n if w_{n}(α) = v(αa) where v denotes the p-adic valuation on ℚ̄_{p}. The set script G sign_{n} of good elements at level n is a subgroup of ℚ̄_{p}*. We prove that each quotient group ℚ̄_{p}*/script G sign_{n} is a torsion group and that each quotient script G sign_{1}/script G sign_{n} is a p-group. We also show that a certain sequence of metric invariants {l _{n}(Z)}_{n∈ℕ} associated to an element Z ∈ B_{dR} ^{+}, is constant.

Original language | English (US) |
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Pages (from-to) | 125-137 |

Number of pages | 13 |

Journal | Kyoto Journal of Mathematics |

Volume | 43 |

Issue number | 1 |

DOIs | |

State | Published - Dec 2003 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

_{dR}

^{+}

*Kyoto Journal of Mathematics*,

*43*(1), 125-137. https://doi.org/10.1215/kjm/1250283743