Formal group law homomorphisms over \(\mathcal O_{\mathbb C_p}\) (Q1043781)

Let \(K\) be a closed discretely valued subfield of \(\mathbb{C}_p\), and \(F,G\) one-dimensional formal group laws of finite height over \(\mathcal{O}_K\). The main result of this paper (Theorem 3.2) shows that if \(\phi:F\rightarrow G\) is a homomorphism defined over \(\mathcal{O}_{\mathbb{C}_p}\) with finite kernel, then there exists a closed discretely valued subfield \(M\subset\mathbb{C}_p\) such that \(\phi\) is defined over \(\mathcal{O}_M\). This provides an analogue of the fact that every homomorphism from \(F\) to \(G\) over \(\mathcal{O}_{K^{\text{alg}}}\) is actually defined over \(\mathcal{O}_L\) for some finite extension \(L|K\), thereby answering a question of \textit{D. J. Schmitz} [New York J. Math. 12, 219--233 (2006; Zbl 1118.14053)]. The first two sections of the paper extend well-known results concerning power-series and formal groups over \(p\)-adic integers to the case where the coefficient ring is \(\mathcal{O}_{\mathbb{C}_p}\). The third and final section provides a proof of Theorem 3.2 and concludes with Example 3.8 showing the failure of this result when \(\ker(\phi)\) is infinite.