Formal groups and $(\phi ,\Gamma )$-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 ${\phi }_E$, 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 étale $({\phi }_E,\Gamma )$-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 ${\phi }_E$ 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 étale $\phi$-modules and continuous representations of the absolute Galois group of the residue field on finitely generated ${\mathcal{O}}_E$-modules.

81 pages. Comments welcome!