The Arithmetic Fundamental Lemma for the Drinfeld space

算术基本引理是算术几何领域中的一类引理，即代数几何和数论的交叉点。它们描述了p-可分群的模空间上的交数和p-进李群上的轨道积分的某些恒等式。特别是，AFL的结构制定了一个p-adic局部领域。它们构成了深层全局断言的局部构建块，这些断言涉及志村簇和自守表示上的循环关系。在这种情况下，它们的作用类似于Langland的基本引理（由Ngô在2009年证明）。AFL猜想的第一个例子是由张伟在2011年发现的。他还证明了低维猜想。从那时起，在局部和全局情况下以及在函数域中，猜想的各种变体已经被制定并部分证明。这些方法介绍了新的方法算术几何，因此，有助于这一领域的数学一般。在这个项目中，我想介绍和研究一个新的变种的AFL猜想的德林费尔德空间。这个变体的一个有趣的方面是退化相交已经在低维中发生。另一方面，Scholze-Weinstein对无限水平的Drinfeld空间作了简单的描述。假设，这可以用来近似地表示所讨论的交叉数。通过这两个方面，该项目有助于更好地理解AFL架构的交集理论方面。麻省理工学院的张伟和云志伟的工作组为该项目的实现提供了理想的环境。我也期待着了解波士顿鼓舞人心的科学氛围。

The arithmetic fundamental lemma conjectures (AFL conjectures) are a family of conjectures in the area of arithmetic geometry, i.e. at the intersection of algebraic geometry and number theory. They describe certain identities of intersection numbers on moduli spaces of p-divisible groups and orbital integrals on p-adic Lie groups. In particular, the AFL conjectures are formulated over a p-adic local field. They form the local building blocks of deep global assertions that concern the relations of cycles on Shimura varieties and automorphic representations. In this way, they take a role which is analogous to Langland's Fundamental Lemma (proven by Ngô in 2009).The first example of an AFL conjecture was found by Wei Zhang in 2011. He also proved the conjecture in low dimension. Various variants of the conjecture have been formulated and partially proven since then, both in local and global situations and over function fields. These approaches introduced new methods to arithmetic geometry and, hence, contributed to this area of mathematics in general.In this project, I would like to introduce and study a new variant of the AFL conjecture for the Drinfeld space. An interesting aspect of this variant is that degenerate intersections occur already in low dimension. Another aspect is that Scholze-Weinstein gave a simple description of the Drinfeld space at infinite level. Presumably, this can be used to approximately express the intersection numbers in question. With both aspects, the project contributes to a better understanding of the intersection theoretic side of the AFL conjectures.The working groups of Wei Zhang and Zhiwei Yun at MIT provide an ideal environment for the realization of this project. I am also looking forward to getting to know the inspiring scientific atmosphere of Boston.