The crystalline period of a height one 𝑝-adic dynamical system

Abstract

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