On the Analyticity of the group action on the Lubin-Tate space

TL;DR

This paper investigates the conditions under which the automorphism group actions on Lubin-Tate spaces are analytic, revealing specific cases where wide open subgroups act analytically on certain discs.

Contribution

It establishes the analyticity of automorphism group actions on Lubin-Tate spaces for particular subgroups and discs, especially involving non-split tori and quasi-canonical liftings.

Findings

Wide open congruence groups of level zero act analytically on certain discs.

Analytic actions are identified on discs related to quasi-canonical liftings.

The period morphism's non-injectivity on some discs is analyzed.

Abstract

In this paper we study the analyticity of the group action of the automorphism group $G$ of a formal module $\bar{F}$ of height 2 (defined over $\overline{\mathbb{F}}_q$) on the Lubin-Tate deformation space $X$ of $\bar{F}$. It is shown that a wide open congruence group of level zero attached to a non-split torus acts analytically on a particular disc in $X$ on which the period morphism is not injective. For certain other discs with larger radii (defined in terms of quasi-canonical liftings) we find wide open rigid analytic groups which act analytically on these discs.