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 $(\phi ,\Gamma )$-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 $\mathrm{Sol}<!--FUNCTION\; APPLICATION-->(\mathcal{ℰ})$ are in a natural correspondence, which gives a geometric interpretation of a result of Berger on $(\phi ,\Gamma )$-modules. In particular, if $V$ is a de Rham Galois representation, its associated filtered $(\phi ,N,G_K)$-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.

我们针对 Fargues-Fontaine 曲线上的等变向量丛开发了 Sen 理论的一个版本。我们证明每个等变向量丛规范地下降到局部解析向量丛。与理论的比较$（\phi ,\gamma ）$然后，分圆情况下的模块恢复 Cherbonnier–Colmez 不完备定理。接下来，我们关注 de Rham 局部解析向量丛的子类。使用$p$-adic monodromy 定理，我们证明每个局部解析向量丛$\mathcal{ℰ}$具有解空间具有满秩的正则微分方程。作为结果，$\mathcal{ℰ}$及其一系列解决方案$\mathrm{索尔}<!--FUNCTION\; APPLICATION-->（\mathcal{ℰ}）$处于自然对应关系，这给出了 Berger 的结果的几何解释$（\phi ,\gamma ）$-模块。特别是，如果$V$是 de Rham Galois 表示，其相关的过滤$（\phi ,氮,G_K）$-模块被实现为微分方程的全局解的空间。我们方法的关键是满足泰特-森形式主义的表示的更高局部分析向量的消失结果，这也是独立的兴趣。