Computations of Orbits for the Lubin-Tate Ring

TL;DR

This paper investigates the action of the automorphism group on Lubin-Tate rings, providing new orbit computations for primes 2 and 3, and confirming known results for larger primes through direct methods.

Contribution

It offers a direct computation approach for the automorphism group orbits on Lubin-Tate rings, including new results for small primes 2 and 3.

Findings

Proved (R_2/p)_{G_2} is isomorphic to F_p for p=2,3

Confirmed known results for p≥5 using different methods

Provided explicit orbit computations for the automorphism group actions

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 $p\geq 5$, the result is a consequence of computations of Shimomura and Yabe and has been reproduced by Kohlhaase using different methods.