What would be the decomposition of $E[p^n]$?

For an elliptic curve (with its algebraic group law ) $$E:\{ (x,y) \in \overline{k}, y^2=x^3+ax+b\}\cup O$$ where $a,b\in k$ is a field of characteristic $0$ then $$E[m] \cong \Bbb{Z}/m\Bbb{Z}\times \Bbb{Z}/m\Bbb{Z}$$ Whence $$E[p^n] \cong \Bbb{Z}_p/p^n\Bbb{Z}_p\times \Bbb{Z}_p/p^n\Bbb{Z}_p$$

• $G[m] \cong \Bbb{Z}/m\Bbb{Z}\times \Bbb{Z}/m\Bbb{Z}$ is obvious for a complex torus $G=\Bbb{C}/\Lambda$

• The Weierstrass functions give some isomorphisms between complex tori and complex elliptic curves

• All the torsion points of $E$ will have their coordinates in $\overline{\Bbb{Q}(a,b)}$, that we can identify with a subfield of $\Bbb{C}$ so it suffices to prove the claim for complex elliptic curves.