Is finiteness of rational points preserved by duality? 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!

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