Kummer‐faithfulness over p‐adic fields

The notion of a Kummer‐faithful field, introduced by Mochizuki, is expected as one of a suitable class of base fields for anabelian geometry. In this paper, we study Kummer‐faithfulness for algebraic extensions of ‐adic fields. We show that Kummer‐faithfulness for such fields are deeply related with various finiteness properties on torsion points of (semi‐) abelian varieties. For example, we prove that a Galois extension of a ‐adic field is Kummer‐faithful with finite residue field if and only if, for every finite extension of and every abelian variety over , the group of ‐rational torsion points is finite. This reveals a close connection between Kummer‐faithfulness and arithmetic finiteness properties of abelian varieties. In addition, using Honda–Tate theory and the local algebraic theory for crystalline characters, we give criteria for Kummer‐faithfulness of Lubin–Tate extension fields.