Modified proof of a local analogue of the Grothendieck conjecture. (English) Zbl 1229.11148

Let \(p\) be a rational prime number, \(E\) a complete discrete valuation field with residue field of characteristic \(p\). Let \(E_{\text{sep}}\) be a fixed separable closure of \(E\) and set \(\Gamma_ E:=\text{Gal}(E_{\text{sep}}/E)\). If \(E(p)\) is the maximal \(p\)–extension of \(E\) in \(E_{\text{sep}}\), let \(\Gamma_ E(p)=\text{Gal}(E(p)/E)\) and let \(\big\{\Gamma_ E(p)^{(\nu)}\big\}_{\nu\geq 0}\) be the upper ramification groups. The main result in this paper is: If \(E\) and \(E'\) are complete discrete valuation fields with finite residue fields and there exists a continuous group isomorphism \(g: \Gamma_ E(p)\longrightarrow \Gamma_{E'}(p)\) such that \(g(\Gamma_ E(p)^ {(\nu)})=\Gamma_ {E'}(p)^{(\nu)}\) for all \(\nu\geq 0\), then there exists a continuous field isomorphism \(\mu : E(p)\longrightarrow E'(p)\) such that \(\mu(E)=E'\) and \(g(\tau)=\mu^{-1}\tau\mu\) for all \(\tau\in \Gamma _ E(p)\).
From this result the corresponding statement for \(E_{\text{sep}}\), \(E'_{\text{sep}}\), \(\Gamma_ E\) and \(\Gamma_{E'}\) follows. This statement is known as a a local analogue of the Grothendieck conjecture.
In [Int. J. Math. 8, No.4, 499–506 (1997; Zbl 0894.11046)] S. Mochizuki proved this local analogue for fields of characteristic \(0\) based on an application of Hodge–Tate theory. The author proved this result for arbitrary characteristic \(p\geq 0\), \(p\neq 2\), in [Int. J. Math. 11, No. 2, 133–175 (2000; Zbl 1073.12501)]. The restriction \(p\neq 2\) is due to the fact that the proof uses the equivalence of \(p\)–groups and the Lie \({\mathbb Z}_ p\)-algebras of nilpotent class \(2\), which holds for \(p>2\). The case \(p=0\) follows from using the theory of fields of norms of Fontaine and Wintenberger.
The proof in the paper under review has no restriction on \(p\) and even though it follows the strategy of the previous paper, there are several essential changes. Firstly, instead of working with the ramification groups \(\Gamma _ E(p)^{(\nu)}\), the author fixes the simplest possible embedding of \(\Gamma_ E(p)\) into its Magnus algebra \({\mathcal A}\) and studies the induced filtration by the ideals \({\mathcal A}^{(\nu)}\), \(\nu\geq 0\), of \({\mathcal A}\). Secondly, any continuous group automorphism of \(\Gamma _ E(p)\) which is compatible with the ramification filtration induces a continuous algebra automorphism \(f\) of \({\mathcal A}\) such that for any \(\nu\geq 0\), \(f({\mathcal A}^{(\nu)}) = {\mathcal A}^{(\nu)}\). This allows to give a more detailed and effective version of the analogue of the Grothendieck conjecture for all \(p>0\). In particular it explains why it holds with the absolute Galois group replaced by the Galois group of the maximal \(p\)–extension.


11S15 Ramification and extension theory
11S20 Galois theory
12F10 Separable extensions, Galois theory
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