Let $f:A\to B$ be a surjective morphism of abelian varieties (over an algebraically closed field).
Why is it true that $f$ induces an epimorphism on the points of finite order $A_{\mathrm{tors}}\to B_{\mathrm{tors}}$?
A surjective morphism of abelian varieties induces an epimorphism on the torsion subgroups
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