The Arithmetic Fundamental Lemma for the Drinfeld space

算术基本引理猜想（AFL猜想）是算术几何领域的一类猜想，即代数几何与数论的交叉。它们描述了p可分群模空间上交点数的恒等式和p进李群上的轨道积分。特别地，AFL猜想是在p进局部场上形成的。它们构成了深层全局断言的局部构件，这些断言涉及志村变量和自同构表示上的循环关系。通过这种方式，它们的作用类似于Langland的基本引理（由Ngô在2009年证明）。第一个AFL猜想的例子是由张伟在2011年发现的。他还在低维上证明了这个猜想。从那时起，这个猜想的各种变体已经在局部和全局情况下以及在函数域上得到了公式化和部分证明。这些方法为等差几何引入了新的方法，因此对这一数学领域做出了贡献。在这个项目中，我想引入和研究德林菲尔德空间的AFL猜想的一个新变体。这种变体的一个有趣的方面是，简并交点已经在低维中出现了。另一方面，Scholze-Weinstein给出了无限层面上的Drinfeld空间的简单描述。据推测，这可以用来近似地表示所讨论的交点数。在这两个方面，该项目有助于更好地理解AFL猜想的交叉理论方面。MIT张伟和恽之伟的工作组为这个项目的实现提供了理想的环境。我也很期待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.