Fisrt, let $k$ be a number field, and let $A/k$ be an abelian variety. I know that by the Mordell–Weil theorem, $A(k)$ is finitely generated, but I saw here (in the introduction) that $A(k)_{tors}$ is finite, but I don't understand why. I tries to search for a proof and I saw that is also true in some other settings:
For example, if $A\to C$ is a family of abelian varieties ($C$ is a projective curve), $A/K$ is an abelian variety for a globalfield $K$, so by Silverman (theorem C.) if the $K(C)/K$-trace of $A$ is zero, then $A(C)_{tors}$ is finite:
I don't understant why both of the claims above are true, I will appreciate any help or hint.