arXiv subjects
Publications and source records attributed to Martin Debaisieux.
We prove Lubin's conjecture for height-one commuting pairs of formal power series defined over the ring of integers $\mathcal{O}$ of any finite extension of $\mathbb{Q}_p$. Using some $p$-adic Hodge theory, we show that such a pair is a pair of endomorphisms of a formal group defined over $\mathcal{O}$.
Let $K$ be a finite extension of $\mathbb{Q}_p$. We prove that a one-dimensional formal group law over $K$ has integral coefficients if and only if its multiplication-by-$n$ endomorphisms have integral coefficients for all integers $n$, in the height-one case, i.e. when the multiplication by $p$ has Weierstrass degree $p$. The proof uses some $p$-adic Hodge theory.
Let $K/\mathbb{Q}_p$ be a finite extension whose ramification index is coprime to $p^2-p$. We study height-one commuting pairs $(f, u)$ of noninvertible and invertible formal power series defined over the ring of integers $\mathcal{O}_K$ of $K$. We begin by extracting a crystalline character of weight $1$ from the $\mathrm{Gal}(\overline K/K)$-set $T_f$ of $f$-consistent sequences. This character is used in order to equip $T_f$ with a $\mathbb{Z}_p$-module structure for which $f$ is an endomorphism. We then apply explicit functors in integral $p$-adic Hodge theory to $T_f$ to recover a formal group defined over $\mathcal{O}_K$ for which $(f, u)$ is a pair of endomorphisms. This proves new cases of a conjecture of Lubin.