Lubin's conjecture for full $p$-adic dynamical systems

TL;DR

This paper proves Lubin's conjecture for full families of commuting $p$-adic power series, showing they originate from endomorphisms of a Lubin-Tate formal group, using ramification and $p$-adic Hodge theory.

Contribution

It provides a concise proof that full commuting $p$-adic power series are endomorphisms of a Lubin-Tate formal group, confirming Lubin's conjecture in this setting.

Findings

Full families of commuting $p$-adic power series are endomorphisms of a Lubin-Tate formal group.

The proof employs ramification theory and $p$-adic Hodge theory techniques.

Lubin's conjecture is validated for large enough families of $p$-adic power series.

Abstract

We give a short proof of a conjecture of Lubin concerning certain families of $p$-adic power series that commute under composition. We prove that if the family is full (large enough), there exists a Lubin-Tate formal group such that all the power series in the family are endomorphisms of this group. The proof uses ramification theory and some $p$-adic Hodge theory.