Kurihara invariants and elimination of wild ramification (Q2151736)

Let \(K\) be a local field of dimension 2 with normalized valuation \(v_K\). Assume that \(K\) has characteristic 0 and residue characteristic \(p\). Then the residue field \(\overline{K}\) of \(K\) is isomorphic to \(K^{(0)}((t))\) for some perfect field \(K^{(0)}\) of characteristic \(p\). Let \(L/K\) be a finite extension and let \(\pi_K\) be a uniformizer for \(K\). The ramification index of \(L/K\) is defined to be \(e_{L/K}=v_L(\pi_K)\). Say that \(L/K\) is unramified if \(e_{L/K}=1\) and the residue field extension \(\overline{L}/\overline{K}\) is separable. The field of constants of \(K\) is defined to be the largest complete subfield \(k\) of \(K\) which has perfect residue field. Say that \(K\) is standard if \(e_{K/k}=1\); in this case \(K\cong k\{\!\{t\}\!\}\) consists of series of the form \(\sum_{n=-\infty}^{\infty}a_nt^n\), where \(a_n\) are elements of \(k\) which satisfy certain valuation conditions. Say that \(K\) is almost standard if there is a finite unramified extension \(L/K\) such that \(L\) is standard. If \(K\) is not almost standard then there exists a finite extension \(\ell/k\) such that \(\ell K\) is almost standard. Assume that \(K\) contains a primitive \(p\)th root of unity, and \(K\) satisfies a certain technical condition (\(K\) is of ``type I''). This paper gives a lower bound on \([\ell:k]\) which is expressed in terms of a rational number \(\Gamma_c(K)\) known as the Kurihara invariant of \(K\). Thus the Kurihara invariant can be used to get explicit lower bounds on how close \(K\) is to being almost standard.