Kummer-faithfulness over $p$-adic fields
Yoshiyasu Ozeki
TL;DR
This work addresses when algebraic extensions of a $p$-adic field are Kummer-faithful, a property tied to the injectivity of Kummer maps for semi-abelian varieties and thus to the feasibility of anabelian geometry over such fields. It develops a general KF framework, then specializes to $p$-adic settings, using Tate curves and $p$-adic Hodge theory to link KF to finiteness of torsion $A(L)_{\mathrm{tor}}$ and $A(L)[\ell^{\infty}]$ under various ramification regimes; it further provides explicit Lubin-Tate extension criteria and constructs non-Kummer-faithful examples. The main contributions are (i) a precise equivalence between KF, AVKF, and torsion-finiteness properties, (ii) criteria for unramified/tamely ramified extensions via residue-field finiteness and quasi-finiteness, and (iii) a detailed, testable criterion for Lubin-Tate extensions along with concrete non-KF instances. These results clarify which base fields are suitable for anabelian methods in the $p$-adic world and illuminate how torsion finiteness phenomena govern Kummer-theoretic injectivity in families of extensions.
Abstract
The notion of a Kummer-faithful field, defined by Mochizuki, is expected as one of suitable base fields for anabelian geometry. In this paper, we study Kummer-faithfulness for algebraic extension fields of $p$-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, a Galois extension $K$ of a $p$-adic field is Kummer-faithful with finite residue field if and only if, for any finite extension $L$ of $K$ and any abelian variety over $L$,its $L$-rational torsion subgroup is finite. In addition, we study Kummer-faithfulness for Lubin-Tate extension fields.