Computations of Orbits for the Lubin-Tate Ring

Abstract

We take a direct approach to computing the orbits for the action of the automorphism group ${\mathbb{G}}_2$ of the Honda formal group law of height $2$ on the associated Lubin-Tate rings $R_2$. We prove that $(R_2/p{)}_{{\mathbb{G}}_2}\cong {\mathbb{F}}_p$. The result is new for $p=2$ and $p=3$. For primes $p\ge 5$, the result is a consequence of computations of Shimomura and Yabe and has been reproduced by Kohlhaase using different methods.