Given a curve $C$. Is there any relation between the etale fundamental group $\pi\_1(C)$ and the first etale cohomology of the constant sheaf , say $Z/nZ$, on $C$ ?


For example, if $C$ is a complex curve, then the singular cohomology $H^1(C,Z)$ is the dual of the topological fundamental group divided by the commutators ( which is the same as Hom$(\pi\_1(C),Z) )$.


So it seems that there should be some relation between Hom$(\pi\_1(C),Z/nZ)$ and $H^1(C,Z/nZ)$ in the etale case, but how?



etale fundamental group and etale cohomology of curves
The two groups you want to compare are canonically isomorphic, so long as C is connected. See Example 11.3 of Milne's notes:


<http://www.jmilne.org/math/CourseNotes/lec.html>


