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

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 paper, first we show a generalization of Imai\u27s result in the case where $A$ has good ordinary reduction. Next we give some finiteness results when $A$ is an elliptic curve and $L$ is the field generated by the $p$-th power torsion of an elliptic curve