Formal moduli spaces of p-divisible groups in residue characteristic 2

我的项目是算术代数几何领域的一部分，更准确地说，它是代数数论、代数几何和志村变分理论的交叉。研究的对象是p可分群的形式模空间。这些模空间由Rapoport和Zink定义。这些空间的一个应用是志村变异的均匀化及其约化的研究，但它们的上同性在局部朗兰对应的背景下也很有趣（参见Fargues， Mantovan和Scholze的工作）。Rapoport和Zink的定义包括EL型模空间（即具有自同态和水平结构形式的额外数据）和PEL型模空间（类似EL，另外具有极化）。残差特征2的排除是该理论的一个重要缺陷。一般定义（独立于残数特征）不仅证明了Rapoport- zink空间理论及其各种（推测的）推广（Rapoport & Viehmann, Kim, Howard & Pappas）的正确性，而且对于在Kudla规划的背景下研究Shimura变体上的特殊循环也是必不可少的。在我的项目中，我想尝试在残差特征2的情况下构造PEL型模空间。这个项目建立在我的博士论文的基础上，在我的博士论文中，我已经定义了一个在分支二次扩展上的两个变量的酉群的模空间。第一个里程碑是将这个结果推广到更多变量的酉群。

My project is part of the field of arithmetic algebraic geometry, more precisely it lies in the intersection of algebraic number theory, algebraic geometry and the theory of Shimura varieties. The objects of study are formal moduli spaces of p-divisible groups. These moduli spaces are defined by Rapoport and Zink. One application of these spaces is the uniformization of Shimura varieties and the study of their reduction, but their cohomology is also interesting in the context of local Langlands correspondences (cp. work of Fargues, Mantovan and Scholze).The definition by Rapoport and Zink includes moduli spaces of type EL (i.e. with extra data in form of endomorphisms and level structures) and of type PEL (like EL, with polarizations in addition) in the case p>2.The exclusion of residue characteristic 2 is a significant gap in this theory. A general definition (independent of the residue characteristic) would not only justify the theory of Rapoport-Zink spaces and their various (conjectural) generalizations (by Rapoport & Viehmann, Kim, Howard & Pappas), it is also essential for the study of special cycles on Shimura varieties in the context of the Kudla program.In my project I want to attempt a construction of such moduli spaces of type PEL in the case of residue characteristic 2. This project builds upon my PhD thesis, where I already defined a moduli spaces for a unitary group in two variables over a ramified quadratic extension. A first milestone would be the generalization of this result to unitary groups in more variables.