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?
You may know the result from complex analysis that for a nonconstant holomorphic function defined on an open set $\Omega \to \mathbb{C}$, the zero set must be discrete. More generally, the preimage of any point of such a function is discrete.
In the scenario of (1), we are dealing with a similar situation, only the mapping is between 2 complex manifolds. But the same result holds: For a holomorphic map between complex manifolds, the preimage of a point is discrete, unless the map is identically constant. That is the result your book is alluding to.
For (2), if a set is discrete, then each point is open, so we have an open cover in which each set in the cover consists of only one point. Now if the set is compact also, then ...