形式群与$(\varphi,\Gamma)$-模
Formal groups and $(φ,Γ)$-modules
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中文总结 AI 辅助
本文通过引入指数周期映射构造环$R_{H,K}$，证明有限生成$\mathcal{O}_E$-模上连续Galois作用与étale $(\varphi_E,\Gamma)$-模的范畴等价，推广了经典分圆与Lubin-Tate情形，并利用$F$-动力系统建立一般等价。
中文摘要 AI 辅助
设$K/E$为$p$-adic局部域的有限非分歧扩张，并设$H$为$\mathcal{O}_K$上有限高的一维形式$\mathcal{O}_E$-模。我们引入$H$的指数周期映射，并利用它构造一个具有不完美剩余域$R_{H,K}$的完全正则局部环、一个自同态$\varphi_E$，以及$\Gamma=\mathrm{Gal}(K(H[p^\infty](\overline{K}))/K)$的交换作用。我们证明了有限生成$\mathcal{O}_E$-模上连续$\mathrm{Gal}_K$-作用与$R_{H,K}$上étale $(\varphi_E,\Gamma)$-模之间的范畴等价。这恢复了相应情形下的经典分圆和Lubin-Tate等价。一般而言，$R_{H,K}$的Krull维数可以大于一，且$\varphi_E$不必提升模$\pi_E$的$q_E$-幂Frobenius。证明使用了$\mathcal{O}_E$上的$F$-动力系统，该系统将模$\pi_E$的收缩与剩余域上的Frobenius相结合。对于每个平坦的$F$-动力系统，我们建立了étale $\varphi$-模与剩余域绝对Galois群在有限生成$\mathcal{O}_E$-模上的连续表示之间的范畴等价。
英文摘要
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 $φ_E$, and a commuting action of $Γ=\mathrm{Gal}(K(H[p^\infty](\overline{K}))/K)$. We prove an equivalence of categories between finitely generated $\mathcal{O}_E$-modules with a continuous $\mathrm{Gal}_K$-action and étale $(φ_E,Γ)$-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 $\mathcal{O}_E$, 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 $\mathcal{O}_E$-modules.
