The Arithmetic Fundamental Lemma for the Drinfeld space

• Applicant: Dr. Andreas Mihatsch
• Subject Area: Mathematics
• Term: from 2019 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 428982207

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.

• DFG Programme: Research Fellowships
• International Connection: USA
• Host: Professor Wei Zhang