Canonical Cohen Rings for Norm Fields

Fix K/Q_p a finite extension and let L/K be an infinite, strictly arithmetically profinite extension in the sense of Fontaine-Wintenberger [24]. Let X_K(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_L/K⁺ ⊂ A⁺ whose reduction modulo p is contained in the valuation ring of X_K(L). When the extension L/K is of a special form, which we call a φ-iterate extension, we prove that X_K(L) is (at worst) a finite purely inseparable extension of Frac A_L/K⁺/pA_L/K⁺). 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.