Semistable models for modular curves and power operations for Morava E-theories of height 2

TL;DR

This paper constructs a semistable integral model for Lubin-Tate curves related to modular curves at supersingular points, enabling a uniform description of power operations in height 2 Morava E-theories.

Contribution

It introduces a new semistable integral model for Lubin-Tate curves and applies it to describe power operations in height 2 Morava E-theories.

Findings

Model is semistable with normal crossing singularities.

Provides a uniform presentation of the Dyer-Lashof algebra.

Connects modular curves with power operations in elliptic cohomology.

Abstract

We construct an integral model for Lubin-Tate curves as moduli of finite subgroups of formal deformations over complete Noetherian local rings. They are p-adic completions of the modular curves X_0(p) at a mod-p supersingular point. Our model is semistable in the sense that the only singularities of its special fiber are normal crossings. Given this model, we obtain a uniform presentation for the Dyer-Lashof algebra of Morava E-theories at height 2 as local moduli of power operations in elliptic cohomology.