Anabelian geometry of complete discrete valuation fields and ramification filtrations

Abstract

As previous studies on anabelian geometry over p-adic local fields suggest, "ramifications of fields" play a key role in this area. In the present paper, more generally, we consider anabelian geometry of complete discrete valuation fields with perfect residue fields from the viewpoint of "ramifications of fields". Concretely, we establish mono-anabelian reconstruction algorithms of various invariants of these fields from their absolute Galois groups with ramification filtrations. By using these results, we reconstruct group-theoretically the isomorphism classes of mixed-characteristic complete discrete valuation fields with perfect residue fields under certain conditions. This result shows that these types of complete discrete valuation fields themselves have some "anabelianness". Moreover, we also investigate properties of homomorphisms between the absolute Galois groups of complete discrete valuation fields with perfect residue fields which preserve ramification filtrations.