Kurihara classification and maximal depth extensions for multidimensional local fields (Q2856437)

Let \(K\) be a local field of dimension \(m\geq2\). Thus there are complete discretely valued fields \(K=K^{(m+1)},K^{(m)},\dots,K^{(1)},K^{(0)}\) such that \(K^{(j)}\) is the residue field of \(K^{(j+1)}\) for \(0\leq j\leq m\). Assume further that char\((K)=0\), char\((K^{(m)})=p>2\), and \(K^{(0)}\) is finite. Let \(k\) denote the algebraic closure of \({\mathbb Q}_p\) in \(K\). Let \(\Omega_{O_K/O_k}\) denote the module of differentials of \(O_K\) over \(O_k\), and let \(\hat{\Omega}_{O_K/O_k}\) denote the completion of \(\Omega_{O_K/O_k}\) with respect to the \(m_K\)-adic topology. Let \(\pi\) be a uniformizer for \(K\), and for \(0\leq j\leq m\) let \(t_j\) be an element of \(O_K\) whose image in \(K^{(j)}\) is a uniformizer for \(K^{(j)}\). Then there are \(a,b_j\in O_K\) such that the \(O_K\)-torsion submodule of \(\hat{\Omega}_{O_K/O_k}\) is generated by \(a\,d\pi+\sum_{j=1}^mb_j\,dt_j\). Define NEWLINE\[NEWLINE\Delta_K(\pi,t_1,\dots,t_m) =\min\{v_K(b_j):1\leq j\leq m\}-v_K(a).NEWLINE\]NEWLINE It is known that \(\Delta_K\) does not depend on the choices of \(a\) and \(b_j\). We say that \(K\) is of type I if \(\Delta_K(\pi,t_1,\dots,t_m)>0\), and of type II if \(\Delta_K(\pi,t_1,\dots,t_m)\leq0\). This classification does not depend on the choices of parameters \(\pi,t_1,\dots,t_m\).NEWLINENEWLINEThis paper contains various results regarding the function \(\Delta_K\). For instance, suppose that \(p\) divides the ramification index \(e_{K/k}\) of \(K/k\). The author proves that \(\Delta_K(\pi,t_1,\dots,t_m)\geq1-v_K(e_{K/k})\), and gives necessary and sufficient conditions for equality. For a complete discretely valued field \(F\) define NEWLINE\[NEWLINEF\{\{X\}\}=\left\{\sum_{n=-\infty}^{\infty} a_nX^n:a_n\in F,\;\inf v_F(a_n)>-\infty,\; \lim_{n\rightarrow-\infty}v_F(a_n)=\infty\right\}.NEWLINE\]NEWLINE Let \(\Delta\) be a positive integer which is not divisible by \(p-1\). It is shown that there is a totally ramified degree-\(p\) extension \(K\) of the ``standard field'' \(K_0=k\{\{X_1\}\}\dots\{\{X_m\}\}\) such that \(\Delta\) is the maximum value achieved by \(\Delta_K(\pi,t_1,\dots,t_m)\) for all choices of parameters \(\pi,t_1,\dots,t_m\) for \(K\). Hence \(K_0\) has infinitely many nonisomorphic degree-\(p\) extensions of type I.