论文标题
Fargues Fontaine曲线上的本地分析矢量束
Locally analytic vector bundles on the Fargues-Fontaine curve
论文作者
论文摘要
在本文中，我们为Fargues-Fontaine Curve上的Equivariant向量捆绑包开发了SEN理论的版本。我们表明，每个模棱两可的矢量束在典型上都降落到本地分析矢量束。与$（φ，γ）$ - 模块的理论进行了比较，然后恢复了Cherbonnier-Colmez年代的变形定理。接下来，我们关注DE RHAM局部分析矢量束的子类别。使用padic monodromy定理，我们表明每个局部分析矢量束$ \ Mathcal {e} $都有一个规范的微分方程，该方程式的解决方案具有完整的等级。结果，$ \ MATHCAL {E} $及其解决方案的捆绑$ \ Mathrm {sol}（\ Mathcal {e}）$自然而然地对应，从而给出了Berger对$（φ，γ）$的几何解释。特别是，如果$ v $是de rham galois表示，则其关联的过滤$（φ，n，g_ {k}）$ - 模块被实现为微分方程全局解决方案的空间。我们方法的一个关键是，对于满足泰特式形式主义的表现形式的较高局部分析向量的消失结果，这也具有独立的利益。
In this article, we develop a version of Sen theory for equivariant vector bundles on the Fargues-Fontaine curve. We show that every equivariant vector bundle canonically descends to a locally analytic vector bundle. A comparison with the theory of $(φ,Γ)$-modules in the cyclotomic case then recovers the Cherbonnier-Colmez decompletion theorem. Next, we focus on the subcategory of de Rham locally analytic vector bundles. Using the p-adic monodromy theorem, we show that each locally analytic vector bundle $\mathcal{E}$ has a canonical differential equation for which the space of solutions has full rank. As a consequence, $\mathcal{E}$ and its sheaf of solutions $\mathrm{Sol}(\mathcal{E})$ are in a natural correspondence, which gives a geometric interpretation of a result of Berger on $(φ,Γ)$-modules. In particular, if $V$ is a de Rham Galois representation, its associated filtered $(φ,N,G_{K})$-module is realized as the space of global solutions to the differential equation. A key to our approach is a vanishing result for the higher locally analytic vectors of representations satisfying the Tate-Sen formalism, which is also of independent interest.
