Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Eigenspaces of Coleman's Trace Operator

Joseph DiCapua

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.

Eigenspaces of Coleman's Trace Operator

Abstract

The Coleman power series defined on a Lubin-Tate tower of extensions over are compatible with respect to two formal group laws: the multiplicative formal group law and some Lubin-Tate formal group law defined over . 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.

Paper Structure

This paper contains 5 sections, 9 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Interpolating $f_0$-norm compatible sequences
  4. The special case $\alpha = q_K/\pi_L^{f_{K/L}}$
  5. Some additional eigenspaces of Coleman's trace operator

Key Result

Lemma 2.0.1

If $h(x) \in \mathscr{D}_{L,K}$ and $\pi_K \mid h(0)$ then $\pi_K \mid h(x)$ in $\mathcal{O}_K[[x]]$.

Theorems & Definitions (9)

  • Lemma 2.0.1
  • Theorem 2.0.2
  • Theorem 2.0.3
  • Theorem 2.0.4
  • Theorem 2.1.1
  • Lemma 2.1.2
  • Lemma 2.1.3
  • Lemma 2.1.4
  • Theorem 3.0.1