TITLE: Torsion points in Abelian varieties over number fields
QUESTION [14 upvotes]: Hello, 
Suppose $A$ is an Abelian variety of dimension $g$ over a number field $k$. Then using height functions one can show that there are non-torsion points in $A(\bar k)$. This looks like an overkill. Is there an easy, elementary way to see this? 
Thanks! 
Ramin

REPLY [5 votes]: Jan Denef once pointed out to me that this is a simple consequence of the Manin-Mumford Conjecture, i.e., Raynaud's Theorem that if $A$ is an Abelian variety defined over a number field and $C$ is a curve on $A$ that is not a coset of an abelian subvariety then $C$ contains only finitely many torsion points.
In this problem we may assume that $A$ has no proper abelian subvarieties.  If $A$ has dimension
at least 2, take $C$ any curve on $A$ defined over $\bar k$, then $C(\bar k)$ is infinite, but 
contains only finitely many torsion points.  If $A$ is an elliptic curve, let $C$ be any curve
of genus at least 2 on $A\times A$. Again, $C(\bar k)$ contains only finitely many torsion points.