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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?

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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

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