An analogue of the Grothendieck conjecture for two-dimensional local fields of finite characteristic.

Abstract

In the case of a local field $K$ of finite characteristic $p>0$, a local analogue of the grothendieck Conjecture appears as a characterization of "analytic" automorphisms of the Galois group $\Gamma _K$ of $K$, i.e. those which are induced by a field automorphism of $K$. Earlier, it was proved by the author that necessary and sufficient conditions for such a characterization in the case of 1-dimensional local fields of characteristic $p\ge 3$ are the compatibility with the ramification filtration of $\Gamma _K$. In the present paper, it is shown that in the case of multidimensional fields, the compatibility with the ramification filtration supplemented by natural topological conditions is still sufficient for the characterization of analytic automorphisms of 4\Gamma _K$.