Isogenies of Riemann surfaces 
*

*Fix $n$ natural. I want to characterize all compact Riemann surfaces $M$ such that $M$ is an unramified covering of degree $n$ over itself.

*How do I construct this covering map?
This map is called an isogeny of $M$.
 A: Following the suggestion of the comment above we can applies Hurwitz's formula, since f is unramified covering we have, 
$$X(M)=nX(M) $$
$n>1$ implies that $X(M)=0$, then  $M$ must be the torus.
Now for the  covering consider the  application, 
$$   (z,w)\in \mathbb{T}\mapsto (z, w^k)\in   \mathbb{T}  $$
is a simple calculation to verify that the application is a covering application.
A: If $f:M\to M$ is a ramified cover of degree $n$ (=non constant morphism= surjective morphism = finite morphism) , Riemann-Hurwitz's formula implies that for the canonical divisor class  $K=K_M$ we have the relation
$$K= f^*K+R$$ where $R$ is the ramification divisor.
Taking degrees and remembering the expression $deg K=2g-2$ for the degree of a canonical divisor in terms of the genus of $g$ of $M$ yields $$   2g-2=n(2g-2)+deg R $$ 
If the covering is known to be unramified (=étale), we have $ deg (R)=0$ (actually even $R=0$) so that $$     2g-2=n(2g-2)        $$  which forces  $n=1$ (duh!) or $g=1$.    
Edit
Conversely, if $g=1$ we have an elliptic curve $M=\mathbb C/\Lambda$ and all of its  holomorphic unramified covers  are of the form $$M\to M:[z]\mapsto [az+b]$$ where $b\in \mathbb C$ is arbitrary and $a\in \mathbb C^*$ is a  complex number satisfying $a\Lambda \subset \Lambda$.
 They are called the isogenies of $M$.
