An intersection number formula for CM cycles in Lubin-Tate towers

TL;DR

This paper derives a comprehensive explicit formula for the intersection numbers of CM cycles in Lubin-Tate spaces, applicable across all levels and quadratic extension cases, with significant implications for arithmetic geometry.

Contribution

It provides the first explicit intersection number formula for CM cycles on Lubin-Tate spaces at all levels, including ramified and unramified cases, and connects to the linear AFL and endomorphism lifting.

Findings

Formula translates the linear AFL into integral comparisons.

Enables recovery of Gross and Keating's endomorphism lifting results.

Works universally for all quadratic extension configurations.

Abstract

We give an explicit formula for the arithmetic intersection number of 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 $K_1,K_2/F$ of non-Archimedean local fields $F$. Our formula works for all cases, $K_1$ and $K_2$ can be either the same or different, ramify or unramified. As applications, this formula translate the linear Arithmetic Fundamental Lemma (linear AFL) into a comparison of integrals. This formula can also be used to recover Gross and Keating's result on lifting endomorphism of formal modules.