It is said that the kernel of a isogeny is finite because it is discrete and complex tori are compact.
I have some questions about this.
1.
Following is my reason for the kernel is discrete.
Suppose $$ \varphi:\mathbb{C}/\Lambda\rightarrow\mathbb{C}/\Lambda' $$
is an isogeny. Then there exists $m\in \mathbb{C}$ such that $m\Lambda=\Lambda'$. So the kernel of $\varphi$ is $\left(\frac{1}{m}\Lambda'\right)/\Lambda$. Intuitively, it is discrete, I think. But I don't know how to reason it.
There is a hint in the book I'm reading saying that if the kernel is not discrete, complex analysis shows that the map is zero.
Can anyone tell me why?
2.
Why can we deduce finiteness from discreteness and compactness?