Geometric Properties of Homomorphisms between the Absolute Galois Groups of Mixed‐characteristic Complete Discrete Valuation Fields with Perfect Residue Fields

Although the analog of the theorem of Neukirch–Uchida for p$p$‐adic local fields fails to hold as it is, a certain analog of this theorem for the absolute Galois groups with ramification filtrations of p$p$‐adic local fields was proved by Mochizuki. Moreover, various necessary and sufficient conditions for homomorphisms between the absolute Galois groups of p$p$‐adic local fields to be “geometric” (i.e., to arise from homomorphisms of fields) were given by Mochizuki and Hoshi. In this paper, we consider similar problems for general mixed‐characteristic complete discrete valuation fields with perfect residue fields and homomorphisms between their absolute Galois groups preserving certain structures related to Hodge–Tate representations. One main result gives necessary and sufficient conditions for homomorphisms between the absolute Galois groups of mixed‐characteristic complete discrete valuation fields with residue fields algebraic over the prime fields to be geometric. We also give a “weak‐Isom” anabelian result (i.e., a sufficient condition for the existence of an isomorphism between two fields in question) for homomorphisms between the absolute Galois groups of these fields.