A generalization of a theorem of Imai and its applications to Iwasawa theory

TL;DR

This paper proves that abelian varieties over certain local fields have finite torsion over infinite p-power extensions, enabling the relaxation of conditions in Iwasawa theory theorems.

Contribution

It generalizes a theorem of Imai to broader fields and demonstrates its applications to Iwasawa theory.

Findings

Finite torsion subgroup for abelian varieties over specified fields

Removal of certain conditions in Iwasawa theory theorems

Extension of Imai's theorem to mixed characteristic fields

Abstract

It is proved that, if $K$ is a complete discrete valuation field of mixed characteristic $(0,p)$ with residue field satisfying a mild condition, then any abelian variety over $K$ with potentially good reduction has finite $K(K^{1/p^\infty})$-rational torsion subgroup. This can be used to remove certain conditions assumed in some theorems in Iwasawa theory.