# Torsion Points of Abelian Varieties with Values in Infinite Extensions over a <i>p</i>-adic Field

### Yoshiyasu Ozeki

Kyoto University, Japan

## 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.

## Cite this article

Yoshiyasu Ozeki, Torsion Points of Abelian Varieties with Values in Infinite Extensions over a <i>p</i>-adic Field. Publ. Res. Inst. Math. Sci. 45 (2009), no. 4, pp. 1011–1031

DOI 10.2977/PRIMS/1260476651