论文标题
多变量（$φ$，$ \ MATHCAL {o} _K^\ times $） - 模块和local-Global兼容性
Multivariable ($φ$,$\mathcal{O}_K^\times$)-modules and local-global compatibility
论文作者
论文摘要
令$ p $为质量数字，$ k $ a $ \ mathbb {q} _p $和$ \ mathbb {f} $ $ \ mathbb {f} _p $的有限扩展。使用PerfectOid空间，我们将$ {\ rm gal}的$ {\ rm gal} $ {\ overline k/k）$ over $ \ mathbb {f} $ aétale$（φ，\ nimcal {o）$ d_a $ d_a^ytlline（$ d_a^ytime）（$ d_a^ytime） $ a $ of $ \ mathbb {f} [\！[\ mathcal {o} _k] \！] $。我们推测，人们还可以将o tale $（φ，\ Mathcal {o} _k^\ times）$ - 模块$ d_a（π）$与任何平稳的表示$ \ Mathrm {gl} _2（k）$ of任何平稳的表示$π$π$π$π$π$π$π$π$π$ \ mathrm {gl} _2（K） $ d_a（π）$是同构（扭曲）至$ d_a^\ otimes（\overlineρ）$，其中$ \overlineρ$是$ {\ rm gal}的基础$ 2 $ 2二维表示（\ overline k/k）$。使用同一位作者的先前工作，我们证明了当$ \overlineρ$是半简单且足够通用时的猜想。
Let $p$ be a prime number, $K$ a finite unramified extension of $\mathbb{Q}_p$ and $\mathbb{F}$ a finite extension of $\mathbb{F}_p$. Using perfectoid spaces we associate to any finite-dimensional continuous representation $\overlineρ$ of ${\rm Gal}(\overline K/K)$ over $\mathbb{F}$ an étale $(φ,\mathcal{O}_K^\times)$-module $D_A^\otimes(\overlineρ)$ over a completed localization $A$ of $\mathbb{F}[\![\mathcal{O}_K]\!]$. We conjecture that one can also associate an étale $(φ,\mathcal{O}_K^\times)$-module $D_A(π)$ to any smooth representation $π$ of $\mathrm{GL}_2(K)$ occurring in some Hecke eigenspace of the mod $p$ cohomology of a Shimura curve, and that moreover $D_A(π)$ is isomorphic (up to twist) to $D_A^\otimes(\overlineρ)$, where $\overlineρ$ is the underlying $2$-dimensional representation of ${\rm Gal}(\overline K/K)$. Using previous work of the same authors, we prove this conjecture when $\overlineρ$ is semi-simple and sufficiently generic.
