Let Hom$(\mathbb{C}/\Lambda_1,\mathbb{C}/\Lambda_2)$ be the set of isogenies between $\mathbb{C}/\Lambda_1$ and $\mathbb{C}/\Lambda_2$, where $\Lambda_1,\Lambda_2$ are lattices.
I am asked to prove what the structure of this group is if:
$\Lambda_1=\mathbb{Z}+\mathbb{Z}i$ and $\Lambda_2=\mathbb{Z}+\mathbb{Z}2i$.
Now I know that if $\psi$ is an isogeny, there exists $\alpha\in\mathbb{C}$ such that $\psi(z\mod\Lambda_1)=\alpha z\mod\Lambda_2$ and $\alpha\Lambda_1\subset\Lambda_2$, and conversely every such $\alpha$ has a corresponding isogeny.
I tried finding the solution as follows: Let $\alpha=a+bi$, let $z=c+di$, where $a,b,c,d\in\mathbb{R}$. Then $\alpha z=ac-bd+(bc+ad)i$. Thus $ac-bd$ has to be an integer multiple of $1$, that is, an integer and $bc+ad$ has to be an integer multiple of 2. If this is correct, how do I continue and if it is not correct, how should it be done?
Note that in your notation, $z$ should be an arbitrary element of the lattice $\Lambda_{1}$ (i.e., $c$ and $d$ are integers), not an arbitrary complex number.
Said another way (eliminating mention of $z$), you should have no trouble showing that $\psi(\Lambda_1)\subseteq\Lambda_2$ if and only if $\alpha\in\Lambda_2$ and $i\alpha\in\Lambda_2$.