sum of torsion of an elliptic curve It is clear from the isomorphism between elliptic curves over $\mathbb{C}$ and complex tori that the sum of the $m$-torsion points is the identity in the group law of the elliptic curve. How generally does this hold, and how can one see it (not using the Lefschetz principle, please)?
 A: It's not true if $m$ is not prime to the characteristic of the field (e.g. take an ordinary elliptic curve in characteristic 2; it will have exactly one non-trivial 2-torsion point).
We also need the field to be algebraically closed, although you may have been assuming that anyway (e.g. take an elliptic curve over $\mathbb{R}$ whose real points have only one connected component - then there's a uniqut non-trivial two-torsion point).
Once we make these two assumptions on the ground field, the torsion is isomorphic to $(\mathbb{Z}/m)^2$ as an abelian group, and this is a property of that group.
A: This more-or-less follows from abstract group theory:
Let $P$ = sum of all points of order $m$.  Then $P$ is itself a point of order $m$.  Now, let $k$ be any integer co-prime to $m$ and note that multiplication by $k$ is a permutation of $E[m]$.  So $[k]P$ = sum of all points in $E[m] = P$.  So $[k-1]P = 0$.  Since $[m]P = 0$ too, then if $gcd( k-1, m ) = 1$ then it follows that $P = 0$.
The case $m = 2$ escapes this proof since $k$ is odd and $gcd( k-1, m ) = 2$.
