Ramification filtrations of certain abelian Lie extensions of local fields (Q301423)

Let \(p\) be a prime and let \(q\) be a power of \(p\). Let \(G\) be a subset of \(\mathbb{F}_q[[x]]\) which forms a compact abelian \(p\)-adic Lie group under the operation of power series substitution. Then there is an APF extension \(L/K\) which corresponds to \(G\) under the field of norms functor. In this paper necessary and sufficient conditions are given on the ramification data of \(G\) for \(K\) to have characteristic 0.NEWLINENEWLINEThese conditions are used in the proof of a special case of Lubin's conjecture from [\textit{J. Lubin}, Compos. Math. 94, No. 3, 321--346 (1994; Zbl 0843.58111), \S6]. Let \(K\) be a finite extension of \(\mathbb{Q}_p\). Assume that there are \(g(x),u(x)\in x\mathcal{O}_K[[x]]\) such that \(g'(0)\in \mathcal{O}_K\smallsetminus\mathcal{O}_K^{\times}\), \(u'(0)\in\mathcal{O}_K^{\times}\), and \(g(u(x))=u(g(x))\). Lubin's conjecture states that, under suitable additional hypotheses, there is a formal group law \(F(x,y)\) over \(\mathcal{O}_K\) such that \(g(x)\) and \(u(x)\) are endomorphisms of \(F\).NEWLINENEWLINENow fix \(g(x)\) as above such that \(g'(0)\) is a uniformizer for \(K\) and \(g(x)\) has Weierstrass degree equal to the size of the residue field of \(\mathcal{O}_K\). Let \(\mathbb{G}\) denote the set of \(u(x)\in x\mathcal{O}_K[[x]]\) such that \(u'(0)\in\mathcal{O}_K^{\times}\) and \(g(u(x))=u(g(x))\). Assume that \(\mathbb{G}\) is full, i.e., \(\{u'(0):u(x)\in\mathbb{G}\}=\mathcal{O}_K^{\times}\). The authors prove that, under these hypotheses, \(g(x)\) is an endomorphism of a Lubin-Tate formal group law defined over \(\mathcal{O}_K\).