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

Abstract: Using the cyclotomic tower, Fontaine introduced $(\varphi,\Gamma)$-modules, which constitute a category equivalent to the category of Galois representations of a $p$-adic local field $K$. In this talk, I will explain a generalization to torsion towers of a one-dimensional formal group $H$. I will first define the exponential period map of $H$ and use it to construct a complete local ring $R_H$ with commuting actions of $\varphi$ and $\Gamma=\mathrm{Gal}(K(H[p^\infty](\overline{K}))/K)$. The main theorem is an equivalence of categories between \'{e}tale $(\varphi,\Gamma)$-modules over $R_H$ and finitely generated $\mathcal{O}_K$-modules with a continuous $\mathrm{Gal}_K$-action, which recovers the classical equivalence in the multiplicative case $H=\widehat{\mathbb{G}}_m$. I will highlight new phenomena which arise in cases of higher height, such as higher Krull dimensions of $R_H$.

algebraic geometrynumber theory

Audience: researchers in the topic

Boston University Number Theory Seminar