Torsion Points of Abelian Varieties with Values in Infinite Extensions over a $p$-adic Field

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 ℤ\_p\_-extension of K. In this paper, first we show a generalization of Imai’s 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.