AN INTERSECTION FORMULA FOR CM CYCLES ON LUBIN-TATE SPACES

We give an explicit formula for the arithmetic intersection number of complex multiplication (CM) cycles on Lubin-Tate spaces for all levels. We prove our formula by formulating the intersection number on the infinite level. Our CM cycles are constructed by choosing two separable quadratic extensions K-1, K-2 over a non-Archimedean local field F. Our formula works for all cases: K-1 and K-2 can be either the same or different, ramified or unramified over F. This formula translates the linear arithmetic fundamental lemma (linear AFL) into a comparison of integrals. As an example, we prove the linear AFL for GL(2)(F).

Automated summary

This article provides an explicit formula for the arithmetic intersection number of complex multiplication (CM) cycles on Lubin-Tate spaces, and proves the formula by formulating the intersection number at the infinite level. The formula works for all cases, whether the extensions are the same or different, and whether they are ramified or unramified over F. Additionally, the article demonstrates the linear arithmetic fundamental lemma for GL(2)(F).