An Analogue of the Grothendieck Conjecture for Two-Dimensional Local Fields of Finite Characteristic

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. the automorphisms of the topological group $\Gamma _K$ induced by conjugation by the automorphisms of the algebraic closure $\overline K$ of $K$ that leave the field $K$ invariant. Earlier, it was proved by the author that necessary and sufficient conditions for such a characterization in the case of one-dimensional local fields $K$ of characteristic $p\geq 3$ are the compatibility of these fields with the ramification filtration of the Galois group $\Gamma _K$. In the present paper, it is shown that, in the case of multidimensional local fields, the compatibility with the ramification filtration supplemented with certain natural topological conditions is still sufficient for the characterization of analytic automorphisms of $\Gamma _K$.