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The action of the Morava stabilizer group on the coefficients of Morava E-theory at height 2

Andrew Salch

TL;DR

The paper solves the explicit action problem for the height $2$ Morava stabilizer group on the Lubin–Tate coefficient ring Def$(bG)$ by converting to Cartier coordinates and deriving a closed, combinatorial formula: the action on the Hazewinkel coordinate $u_1$ is a weighted sum over $p$-labelled ordered rooted trees, with coefficients given by tree indices; it further provides a closed description of the action on the graded ring Def$_*(bG)\

Abstract

We calculate an explicit closed formula for the action of the height 2 full Morava stabilizer group on the coefficient ring of height 2 Morava E-theory. In particular, this yields an explicit, surprisingly simple closed formula for the action of the automorphism group of a height 2 formal group law on its Lubin-Tate deformation ring. The formula is of a combinatorial nature, given by sums over certain labelled ordered rooted trees.

The action of the Morava stabilizer group on the coefficients of Morava E-theory at height 2

TL;DR

The paper solves the explicit action problem for the height Morava stabilizer group on the Lubin–Tate coefficient ring Def by converting to Cartier coordinates and deriving a closed, combinatorial formula: the action on the Hazewinkel coordinate is a weighted sum over -labelled ordered rooted trees, with coefficients given by tree indices; it further provides a closed description of the action on the graded ring Def$_*(bG)\

Abstract

We calculate an explicit closed formula for the action of the height 2 full Morava stabilizer group on the coefficient ring of height 2 Morava E-theory. In particular, this yields an explicit, surprisingly simple closed formula for the action of the automorphism group of a height 2 formal group law on its Lubin-Tate deformation ring. The formula is of a combinatorial nature, given by sums over certain labelled ordered rooted trees.

Paper Structure

This paper contains 16 sections, 14 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. History of the problem
  3. Motivations for the problem
  4. The main results
  5. Conventions
  6. Acknowledgments
  7. Review of Devinatz, Gross, and Hopkins' use of Cartier theory at height $2$
  8. From Hazewinkel coordinates to Cartier coordinates
  9. The recursive formula for the action of ${\rm Aut}(\mathbb{G})$ on $u_1$
  10. The closed formula for the action of ${\rm Aut}(\mathbb{G})$ on $u_1$
  11. The action of $W(\mathbb{F}_{q^2})^\times \subseteq {\rm Aut}(\mathbb{G})$ on $u_1$
  12. The action of ${\rm Aut}(\mathbb{G})$ and of $W(\mathbb{F}_{q^2})$ on the Bott element $u$
  13. Examples of ordered rooted $q$-labelled trees
  14. Weights 1 and 2
  15. Weight 3
  16. ...and 1 more sections

Key Result

Theorem A

Let $\gamma_n\in W(\mathbb{F}_{p^2})$ be the coefficient of $u_1^n$ in the power series $(\alpha_0 + \alpha_1S).u_1$, i.e., Then $\gamma_n$ is the sum, over all $p$-labelled ordered rooted trees $T$ of weight $n$, of the $(\alpha_0,\alpha_1)$-index of $T$:

Theorems & Definitions (31)

  • Theorem A
  • Theorem B
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • Proposition 4.1
  • proof : Proof of Proposition \ref{['u1 formula']}
  • Definition 5.1
  • Definition 5.2
  • ...and 21 more