### Abstract

Fix K/Q<inf>p</inf> a finite extension and let L/K be an infinite, strictly arithmetically profinite extension in the sense of Fontaine-Wintenberger [24]. Let X<inf>K</inf>(L) denote its associated norm field. The goal of this paper is to associate to L/K, in a canonical and functorial way, a p-adically complete subring A<inf>L/K</inf><sup>+</sup> ⊂ A<sup>+</sup> whose reduction modulo p is contained in the valuation ring of X<inf>K</inf>(L). When the extension L/K is of a special form, which we call a φ-iterate extension, we prove that X<inf>K</inf>(L) is (at worst) a finite purely inseparable extension of Frac A<inf>L/K</inf><sup>+</sup>/pA<inf>L/K</inf><sup>+</sup>). The class of φ-iterate extensions includes all Lubin-Tate extensions, as well as many other extensions such as the non-Galois "Kummer" extension occurring in the work of Faltings, Breuil, and Kisin. In particular, our work provides a canonical and functorial construction of every characteristic zero lift of the norm fields that have thus far played a foundational role in (integral) p-adic Hodge theory, as well as many other cases which have yet to be studied.

Original language | English (US) |
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Pages (from-to) | 5473-5517 |

Number of pages | 45 |

Journal | International Mathematics Research Notices |

Volume | 2015 |

Issue number | 14 |

DOIs | |

State | Published - 2015 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*International Mathematics Research Notices*,

*2015*(14), 5473-5517. https://doi.org/10.1093/imrn/rnu098