Locally analytic vector bundles on the Fargues–Fontaine curve

IF 0.9 1区 数学 Q2 MATHEMATICS
Gal Porat
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引用次数: 0

Abstract

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 has a canonical differential equation for which the space of solutions has full rank. As a consequence, and its sheaf of solutions Sol () 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,GK)-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.

法尔古斯-方丹曲线上的局部解析向量束
我们为法尔古斯-方丹曲线上的等变向量束建立了一个森理论版本。我们证明了每一个等变向量束都能典型地降到一个局部解析向量束。通过与循环情况下的(φ,Γ)模块理论进行比较,我们发现了谢邦尼尔-科尔梅兹反完备性定理。接下来,我们关注 de Rham 局部解析向量束子类。利用 p-adic 单调性定理,我们证明了每个局部解析向量束 ℰ 都有一个典范微分方程,其解的空间具有全秩。因此,ℰ 和它的解组 Sol (ℰ) 是自然对应的,这就给出了伯杰关于 (φ,Γ) 模块的一个结果的几何解释。特别是,如果 V 是一个 de Rham 伽罗瓦表示,那么它的相关滤波 (φ,N,GK) 模块就是微分方程全局解的空间。我们方法的关键是满足塔特-森形式主义的表示的高局部解析向量的消失结果,这也是我们的兴趣所在。
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来源期刊
CiteScore
1.80
自引率
7.70%
发文量
52
审稿时长
6-12 weeks
期刊介绍: ANT’s inclusive definition of algebra and number theory allows it to print research covering a wide range of subtopics, including algebraic and arithmetic geometry. ANT publishes high-quality articles of interest to a broad readership, at a level surpassing all but the top four or five mathematics journals. It exists in both print and electronic forms. The policies of ANT are set by the editorial board — a group of working mathematicians — rather than by a profit-oriented company, so they will remain friendly to mathematicians'' interests. In particular, they will promote broad dissemination, easy electronic access, and permissive use of content to the greatest extent compatible with survival of the journal. All electronic content becomes free and open access 5 years after publication.
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