The crystalline period of a height one $p$-adic dynamical system over $\mathbf{Z}_p$

TL;DR

This paper proves that certain $p$-adic dynamical systems over $Z_p$ with specific properties are necessarily endomorphisms of formal groups, confirming a case of Lubin's conjecture using Fontaine's crystalline period ring.

Contribution

It establishes that height one $p$-adic dynamical systems with specified conditions are endomorphisms of formal groups, verifying a case of Lubin's conjecture over $Z_p$.

Findings

Such systems are formal group endomorphisms.

The proof uses Fontaine's crystalline period ring embedding.

It confirms a height one case of Lubin's conjecture.

Abstract

Let $f$ be a continuous ring endomorphism of $\mathbf{Z}_p[[x]]/\mathbf{Z}_p$ of degree $p.$ We prove that if $f$ acts on the tangent space at $0$ by a uniformizer and commutes with an automorphism of infinite order, then it is necessarily an endomorphism of a formal group over $\mathbf{Z}_p.$ The proof relies on finding a stable embedding of $\mathbf{Z}_p[[x]]$ in Fontaine's crystalline period ring with the property that $f$ appears in the monoid of endomorphisms generated by the Galois group of $\mathbf{Q}_p$ and crystalline Frobenius. Our result verifies, over $\mathbf{Z}_p,$ the height one case of a conjecture by Lubin.