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

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.