The Guo-Jacquet fundamental lemma is a higher dimensional generalization of the local field analogue of the Waldpurger formula. Its arithmetic generalization is the linear arithmetic fundamental lemma conjectured by Wei Zhang. It interprets the derivative of certain orbital integral in terms of intersection numbers on Lubin-Tate spaces, and is a local analogue of the Gross-Zagier formula. We will introduce the known results for the linear fundamental lemma, and intersection numbers on Lubin-Tate spaces. After a joint work with Ben Howard, we also discovered a bi-quadratic generalization of the conjecture.

Title: Linear Arithmetic Fundamental Lemma and Intersection Numbers for CM cycles on Lubin-Tate spaces