On Gross-Keating's result of lifting endomorphisms for formal modules

TL;DR

This paper provides an alternative proof to Gross and Keating's result on lifting endomorphisms of formal modules, using intersection formulas of CM cycles in Lubin-Tate spaces, instead of formal cohomology theory.

Contribution

It offers a new proof of Gross-Keating's theorem on endomorphism lifting, utilizing intersection theory in deformation spaces.

Findings

Explicit determination of endomorphism rings between  s and  D.

Alternative proof using intersection formulas of CM cycles.

Connections between formal module endomorphisms and deformation space geometry.

Abstract

$\newcommand{\OO}[1]{\mathcal{O}_{#1}}\newcommand{\GG}{\mathcal{G}}\newcommand{\End}{\mathrm{End}}\newcommand{\O}{\mathcal{O}}$Let $K/F$ be a quadratic extension of non-Archimedean local fields of characteristic not equal to 2, with rings of integers denoted by $\OO K$ and $\OO F$. We consider a formal $\OO F$-module $\GG$, over a discrete valuation ring $\OO W$ with an uniformizer $\varpi$, with extra endomorphisms by a subring $\O$ of $\OO K$, and the height of its reduction $\GG_0=\GG\otimes \OO W/\varpi$ is 2. The endomorphism ring of $\GG_n=\GG\otimes \OO W/\varpi^{n+1}$ is a subring between $\OO s$ and $\OO D=\End(\GG_0)$. We will determine them explicitly. This result was previously proved by Gross and Keating. Their treatment is the formal cohomology theory. We will give another proof using the intersection formula of CM cycles in Lubin-Tate deformation spaces.