Elliptic Curves

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 $q=charK$.

Fact: If $m$ is coprime to $q$ then $\mid E[m]\mid =m^2$, 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 $v>0$.

We can combine the above results (using the fact that $E[\mathrm{ab}]=E[a]\times E[b]$ 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 $E[m]\cong {\mathbb{Z}}_n\times {\mathbb{Z}}_m$