Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 18 / 1 / i

Let $E=\mathbb{C}/\Lambda$, and let ${\Lambda }^{\prime }$ be the inverse image of the cyclic subgroup $G$ under the quotient map. Then $\Lambda \subset {\Lambda }^{\prime }$ is an inclusion of Euclidean lattices with ${\Lambda }^{\prime }/\Lambda \cong \mathbb{Z}/N\mathbb{Z}$. The Smith normal form supplies a basis $u,v$ of ${\Lambda }^{\prime }$ for whichChoose the orientation of this basis so that $\mathrm{Im}(v/(Nu))>0$. Dividing the coordinate on $\mathbb{C}$ by $Nu$ gives an isomorphism of complex toriHere $u$ maps to $1/N$, and its class generates ${\Lambda }^{\prime }/\Lambda$. Thus the isomorphism carries $G$ to the required cyclic subgroup. Every pair is represented: $\Phi$ is surjective. This argument avoids assuming that an arbitrary integer lift of a primitive vector modulo $N$ is itself primitive.