Torsion Points

Consider the multiplication-by-\(m\) map \([m]\). Then the group of all points \(P\) such that \([m] P = O\) is denoted \(E[m]\). [My notes are unclear here, but I think we are now working over the algebraic closure of \(K\).]

Let \(q = \mathrm{char} K\).

Fact: If \(m\) is coprime to \(q\) then \(|E[m]| = m^2\), and furthermore \(E[m] \cong \mathbb{Z}_m \times \mathbb{Z}_m\).

Fact: If \(E[q] \ne \{O\}\) then \(E[q^v] \cong \mathbb{Z}_{q^v}\) for all \(v \gt 0\).

We can combine the above results (using the fact that \(E[ab] = E[a] \times E[b]\) for coprime \(a,b\). Let \(m\) be a positive integer. Write \(m = q^v n\) where \(q\) does not divide \(n\).

  • If \(E[q] = \{O\}\) we have \(E[m] \cong \mathbb{Z}_n \times \mathbb{Z}_n\)

  • Otherwise \(E[m] \cong \mathbb{Z}_n \times \mathbb{Z}_m\)


Ben Lynn blynn@cs.stanford.edu 💡