Formal groups and $(\varphi,\Gamma)$-modules

Abstract

Let $K/E$ be a finite unramified extension of $p$-adic local fields, and let $H$ be a one-dimensional formal $\mathcal{O}_E$-module of finite height over $\mathcal{O}_K$. 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 $\varphi_E$, and a commuting action of $\Gamma=\mathrm{Gal}(K(H[p^\infty](\overline{K}))/K)$. We prove an equivalence of categories between finitely generated $\mathcal{O}_E$-modules with a continuous $\mathrm{Gal}_K$-action and \'{e}tale $(\varphi_E,\Gamma)$-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 $\varphi_E$ need not lift the $q_E$-power Frobenius modulo $\pi_E$. The proof uses $F$-dynamical systems over $\mathcal{O}_E$, which combine contraction modulo $\pi_E$ with Frobenius on the residue field. For every flat $F$-dynamical system, we establish an equivalence of categories between \'{e}tale $\varphi$-modules and continuous representations of the absolute Galois group of the residue field on finitely generated $\mathcal{O}_E$-modules.