Elementary computation of the stable reduction of the Drinfeld modular curve $X(\pi^2)$

TL;DR

This paper explicitly constructs the stable reduction of the Drinfeld modular curve $X(^2)$ in equal characteristic, providing a detailed stable covering and intersection data, based on elementary methods inspired by Coleman-McMurdy.

Contribution

It offers a new explicit and elementary method to compute the stable reduction of $X(^2)$, including a stable covering and intersection multiplicities.

Findings

Constructed a stable covering of $X(^2)$

Computed intersection multiplicity data

Provided an explicit elementary approach

Abstract

We compute the stable reduction of the Lubin-Tate space $X(\pi^2)$ in the equal characteristic case, on the basis of Coleman-McMurdy's ideas. Namely, in this paper, we actually construct a stable covering of $X(\pi^2).$ This paper also includes intersection multiplicity datum of the stable model of $X(\pi^2)$. Our method is very explicit and elementary.