Higher criteria for the regularity of a one-dimensional local field (Q2674745)

Let \(K/\mathbb Q_p\) be a finite extension of local fields, \(\mathcal O\) the valuation ring of \(K\), \(\mathcal M\) the valuation ideal of \(\mathcal O\), and \(F \in \mathcal O [[X,Y]]\) a formal group (law) of height \(h\). Then the ring of endomorphisms of \(F(\mathcal M)\) is isomorphic to the valuation ring \(\mathcal O_{K_0}\) of some subfield \(K_0 \subset K\). For each \(a \in \mathcal O_{K_0}\) let \([a]\) denote the endomorphism under this isomorphism. Choose a uniformizer \(\pi_0\) of \(K_0\), and let \(q\) denote the cardinality of the residue class field of \(\mathcal O_{K_0}\). The notion of (ir)regular local field is extended to formal groups as follows: an extension field \(L/K\) (or more precisely, the \(\mathcal O_{K_0}\)-module \(F(\mathcal M_L)\), where \(\mathcal M_L\) is the valuation ideal in \(L\)) is called \textit{regular}, if \(L\) does not contain \(\ker [\pi_0]\); it is called \textit{\(s\)-irregular}, if \(L\) does not contain \(\ker [\pi_0^{s+1}]\) for some \(1 \le s \in \mathbb N\); and it is called \textit{completely regular} (resp. \textit{completely \(s\)-irregular}), if every unramified extension \(M/L\) has this property. Supposing that \(K/K_0\) is unramified, it is shown in Theorem 2 that \(L/K\) is completely regular if the ramification index \(e_{L/K}\) is not divisible by \(q^h-1\). In the rest of the paper the authors present criteria for the \(s\)-irregularity of \(L/K\) presupposing that \(F\) has height \(h=1\) and \(K/K_0\) is abelian. The main results (for an arbitrary formal group \(F\)) are given in Theorems 3 and 4, and in the following sections these results are applied to Lubin-Tate groups (here \(K_0=K\)), to polynomial formal groups (here \(K_0=\mathbb Q_p\)), to relative Lubin-Tate groups and to generalized Lubin-Tate groups.