Isogenies between dynamics of formal groups
The paper at hand is the continuation of research of the author about formal series over the ring of integers \(\mathcal O\) of a \(\mathfrak p\)-adic number field \(K\) (see \textit{H. C. Li} [Compos. Math. 100, 351-364 (1996; Zbl 0871.11084)]). This research is motivated by a vague conjecture of J. Lubin, that ``for an invertible series to commute with a noninvertible series, there must be a formal group somehow in the background.'' Let \(f(x)\in{\mathcal O}x\) be an endomorphism of a formal group, let \(\overline{\mathcal M}\) be the ring of integers in the algebraic closure \(\overline{K}\) of \(K\), and let \(\Lambda_n(f):=\{\alpha\in\overline{{\mathcal M}}\mid f^{\circ n}(\alpha)= 0\}\), \(\Lambda=\bigcup^\infty_{n=1} \Lambda_n(f)\). The formal group induces a group structure on \(\Lambda(f)\). The author proves that the roots in \(\overline{{\mathcal M}}\) of an \(\mathcal O\)-isogeny of \(f(x)\) form a subgroup of \(\Lambda(f)\). The author considers also some generalizations to the case that \(f(x)\) is not an endomorphism of a formal group.
scientific article; zbMATH DE number 992061