Formal groups and (φ,Γ)-modules

@misc{kiyohara2026formal,
  title = {{Formal groups and $(φ,Γ)$-modules}},
  author = {Daishi Kiyohara},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31608},
  eprint = {2609.31608},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31608},
}

TY  - UNPB
TI  - Formal groups and (φ,Γ)-modules
AU  - Daishi Kiyohara
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.31608
UR  - https://arxiv.org/abs/2609.31608
AB  - Let K/E be a finite unramified extension of p-adic local fields, and let H be a one-dimensional formal 𝒪_E-module of finite height over 𝒪_K. We introduce the exponential period map of H and use it to construct a complete regular local ring R_H,K with imperfect residue field, an endomorphism φ_E, and a commuting action of Γ=Gal(K(H[p^∞](K̅))/K). We prove an equivalence of categories between finitely generated 𝒪_E-modules with a continuous Gal_K-action and étale (φ_E,Γ)-modules over R_H,K. This recovers the classical cyclotomic and Lubin-Tate equivalences in the corresponding cases. In general, R_H,K can have Krull dimension greater than one, and φ_E need not lift the q_E-power Frobenius modulo π_E. The proof uses F-dynamical systems over 𝒪_E, which combine contraction modulo π_E with Frobenius on the residue field. For every flat F-dynamical system, we establish an equivalence of categories between étale φ-modules and continuous representations of the absolute Galois group of the residue field on finitely generated 𝒪_E-modules.
ER  -