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The following statement in a paper puzzles me:

"We may view $H^1(X(N), \mathbb{Z}/\ell\mathbb{Z})$ as classifying unramified Galois coverings of $X(N)$ with structure group $\mathbb{Z}/\ell\mathbb{Z}$."

Here $X(N)$ is the usual modular curve of level $N$. Anyway, the statements seems to indicates that it is a general result about (complete?) curves over $\mathbb{C}$.

On one hand, the ├ętale covers of a variety corresponds to finite continuous $\pi_1^{\mathrm{ét}}(X)$-sets. On the other hand, $H^1(X(N), \mathbb{Z}/\ell\mathbb{Z})\cong (\mathbb{Z}/\ell\mathbb{Z})^{2g}$ is identified with the $n$-torsions of the Picard variety through the Kummer sequence. I am having trouble to tie these two sides of the picture together. Could any explain or point to a nice reference? Thank you.

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7  
In topology, Galois coverings of a (nice, path-connected) space $X$ with structure group $G$ are classified by morphisms $\pi_1(X,x_0)\to G$. If $G$ is abelian then such morphisms are the same as elements of $H^1(X,G)$ (by Hurewicz homomorphism and universal coefficients theorem). Your question is therefore: why are etale/algebraic definitions the same as topological, in the case of finite sets/groups? I suppose you know the answer very well (better than me) [a little piece: every finite (topological) cover of a curve $X$ is algebraic by existence of merom. functions / Riemann-Roch] –  user8268 Jun 9 '11 at 8:48
    
@user8268, Thanks a lot. I get it now. –  Jiangwei Xue Jun 9 '11 at 9:57
4  
Dear Jiangwei, You have good taste in reading material! Regards, –  Matt E Jun 9 '11 at 10:50
    
feels so happy to see a teacher appreciating a student for his hardwork :) –  user9413 Jun 13 '11 at 19:08

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