Is finiteness of rational points preserved by duality?

Question

Sorry if this is obvious. I don't know much about Abelian varieties.

Let $A/k$ be an abelian variety. Let's say $k$ has characteristic zero.

Let $\widehat{A}$ be the dual abelian variety.

Suppose that the set $A(k)$ of $k$-rational points is finite.

Is $\widehat{A}(k)$ also finite?

References will be much appreciated!

Answer 1

Since any abelian variety is isogenous to its dual, the answer is yes.