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

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.