= Solution
Let $E=\mathbb C/\Lambda$, and let $\Lambda'$ be the inverse image of the <cyclic subgroup> $G$ under the quotient map. Then $\Lambda\subset\Lambda'$ is an inclusion of <Euclidean lattices> with $\Lambda'/\Lambda\cong\mathbb Z/N\mathbb Z$. The <Smith normal form> supplies a basis $u,v$ of $\Lambda'$ for which
$$
\Lambda=\mathbb Z(Nu)+\mathbb Zv.
$$
Choose the orientation of this basis so that $\operatorname{Im}(v/(Nu))>0$. Dividing the coordinate on $\mathbb C$ by $Nu$ gives an isomorphism of <complex tori>
$$
E\cong \mathbb C/(\mathbb Z+\mathbb Z\tau),\qquad \tau=\frac{v}{Nu}.
$$
Here $u$ maps to $1/N$, and its class generates $\Lambda'/\Lambda$. Thus the isomorphism carries $G$ to the required <cyclic subgroup>. \b[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.
Back to article page
