sum of torsion of an elliptic curve

Question

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)?

Answer 1

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.

Answer 2

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 $gcd( k-1, m ) = 2$.