Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
up vote 1 down vote accepted

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 $i\alpha\in\Lambda_2$.

share|cite|improve this answer
I agree, $c,d\in\mathbb{R}$, so then from $bc+ad$ is even, it follows that $b$ and $d$ are even and from $ac-bd$ integer, it follows that $b,d$ are integers. But wouldn't then $\psi(\Lambda_1)\subseteq\Lambda_2$ iff $\alpha=2\mathbb{Z}+2\mathbb{Z}i$. This is equivalent to saying $\alpha\in\Lambda_2$ and $i\alpha\in\Lambda_2$. But is there an easier way to see your answer immediately? – user100659 Oct 18 '13 at 13:28
And in case $\Lambda_1=\mathbb{Z}+\mathbb{Z}i$ and $\Lambda_2=\mathbb{Z}+\mathbb{Z}\sqrt{-2}$. Would we get $ac-bd$ integer and $bc+ad$ a multiple of $\sqrt{2}$, which seems impossible so the group Hom will be empty? – user100659 Oct 18 '13 at 13:37
To expand my second observation, since $1$ and $i$ generate $\Lambda_1$ (and $\Lambda_2$ is a lattice), $\psi(\Lambda_1)\subseteq\Lambda_2$ if and only if $\alpha=\psi(1)\in\Lambda_2$ and $i\alpha=\psi(i)\in\Lambda_2)$. (And "yes" to your question about $\Lambda_2=\mathbf{Z}+\mathbf{Z}\sqrt{-2}$. :) – Andrew D. Hwang Oct 18 '13 at 13:40
Awesome, thanks! – user100659 Oct 18 '13 at 13:45

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.