Lubin's conjecture for height-one $p$-adic dynamical systems over $p(p-1)$-tame extensions

Abstract

We prove Lubin's conjecture for height-one commuting pairs of formal power series defined over the ring of integers $\mathcal{O}$ of finite extensions of ${\mathbb{Q}}_p$ whose ramification index is coprime to $p(p-1)$. For such a pair $(f,u)$, with $f$ noninvertible and $u$ invertible, we show that $(f,u)$ is a pair of endomorphisms of a formal group defined over $\mathcal{O}$. The proof relies on extracting a crystalline character of weight $1$ from the Galois set $T_f$ of $f$-consistent sequences, using it to define a ${\mathbb{Z}}_p$-module structure on $T_f$ for which $f$ is an endomorphism, and applying explicit functors in integral $p$-adic Hodge theory to recover the latent formal group.

arXiv comment17 pages, 1 figure