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，更确切地说，它是在交叉的代数数论，代数几何和理论的志村品种。研究对象是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簇的一致化及其约化的研究，但它们的上同调在局部朗兰兹对应的上下文中也很有趣（cp. Rapoport和Zink的定义包括EL型模空间（即具有以自同态和能级结构形式的额外数据）和PEL型模空间（类似EL，具有额外的极化）（p> 2）。一般定义（独立于剩余特征）不仅证明了Rapoport-Zink空间及其各种理论的合理性，（语法）概括（作者：Rapoport & Viehmann，Kim，霍华德& Pappas），在Kudla计划的背景下，它对于研究Shimura簇上的特殊圈也是必不可少的。在我的项目中，我想尝试在以下情况下构造这样的PEL型模空间：残基特性2.这个项目建立在我的博士论文，在那里我已经定义了一个模空间的酉群在两个变量在一个分歧的二次扩张。第一个里程碑是将这一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.