Eigenspaces of Coleman's Trace Operator

TL;DR

This paper investigates the eigenspaces of Coleman's trace operator to understand the compatibility of power series with respect to multiple Lubin-Tate formal group laws, extending previous work on Coleman power series.

Contribution

It introduces a classification framework for power series compatible with two Lubin-Tate formal group laws using eigenspaces of Coleman's trace operator.

Findings

Partial classification of such power series.

Introduction of new eigenspaces of Coleman's trace operator.

Insights into compatibility conditions for formal group laws.

Abstract

The Coleman power series defined on a Lubin-Tate tower of extensions over $K$ are compatible with respect to two formal group laws: the multiplicative formal group law and some Lubin-Tate formal group law defined over $\mathcal{O}_K$. We ask if it is possible to generalize these power series in order to find power series which are compatible with respect to two Lubin-Tate formal group laws in the same way. We provide a precise formulation of this question and a partial answer towards the classification of all such power series which involves the eigenspaces of Coleman's trace operator. Some additional eigenspaces of Coleman's trace operator are also introduced.