Abstract : Let A be an Abelian variety over a field K. The group A(K) of K-rational points on A, known as the Mordell-Weil group of A, is known to be finitely generated if K is an algebraic number field of finite degree. It is known to be of infinite rank if K is a certain type of algebraic number field of infinite degree. If K is "too large", then A(K) contains a non-trivial divisible subgroup. I will discuss some reasonable conditions on K which allow A(K) to contain no non-trivial divisible subgroups, and give some examples of such K. |