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

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.

16 pages