Torsion points of abelian varieties with values in infinite extensions over a p-adic field

TL;DR

This paper generalizes finiteness results for torsion points of abelian varieties over infinite extensions of p-adic fields, extending Imai's theorem to broader cases including ordinary good reduction.

Contribution

It extends Imai's finiteness theorem to abelian varieties with ordinary good reduction and explores finiteness in elliptic curve torsion points over specific infinite extensions.

Findings

Finiteness of torsion points for abelian varieties with ordinary good reduction.

Finiteness results for elliptic curves over fields generated by p-power torsion.

Generalization of Imai's theorem beyond cyclotomic extensions.

Abstract

Let $A$ be an abelian variety over a $p$-adic field $K$ and $L$ an algebraic infinite extension over $K$. We consider the finiteness of the torsion part of the group of rational points $A(L)$ under some assumptions. In 1975, Hideo Imai proved that such a group is finite if $A$ has good reduction and $L$ is the cyclotomic $\mathbb{Z}_p$-extension of $K$. In this talk, first we show a generalization of Imai's result in the case where $A$ has ordinary good reduction. Next we give some finiteness results when $A$ is an elliptic curve and $L$ is the field generated by the $p$-power torsion of an elliptic curve.