Constructing $2$-dimensional Lubin-Tate formal groups over $\mathbb{Z}_{p}$ (I)

TL;DR

This paper constructs 2-dimensional Lubin-Tate formal groups over 9p and explores the abelian extensions generated by their torsion points, revealing ramification properties and generalizations of classical 1-dimensional theory.

Contribution

It introduces a new class of 2-dimensional formal groups over 9p and studies their torsion points and associated abelian extensions, extending Lubin-Tate theory.

Findings

Coordinates of p^9-torsion points generate abelian extensions.

The extension from p-torsion points is generally totally ramified.

The work generalizes classical Lubin-Tate formal groups to higher dimensions.

Abstract

In this paper, we construct a class of $2$-dimensional formal groups over $\mathbb{Z}_p$ that provide a higher-dimensional analogue of the usual $1$-dimensional Lubin-Tate formal groups, then we initiate the study of the extensions generated by their $p^{n}$-torsion points. For instance, we prove that the coordinates of the $p^{\infty}$-torsion points of such a formal group generate an abelian extension over a certain unramified extension of $\mathbb{Q}_{p}$, and we study some ramification properties of these abelian extensions. In particular, we prove that the extension generated by the coordinates of the $p$-torsion points is in general totally ramified.