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 étale 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.