Let $X$ be a proper smooth curve over a field $k$. Then we have an exact sequence of profinite groups \begin{equation*} 1 \to \pi_1(X_{\overline k}) \to \pi_1(X) \to G_k \to 1, \end{equation*} which induces an outer Galois representation \begin{equation*} G_k \to \mathop{\mathrm{Out}}\left(\pi_1(X_{\overline k})\right). \end{equation*} Show that after abelianization, we have an isomorphism of $G_k$modules \begin{equation*} \pi_1(X_{\overline k})^\mathrm{ab} \cong \prod_\ell T_\ell(J_X), \end{equation*} where $J_X$ denotes the Jacobian variety of $X$ and the $G_k$module structure on $\pi_1(X_{\overline k})^\mathrm{ab}$ is induced by the outer Galois representation.
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2$\begingroup$ Have you tried looking in Szamuely's book "Galois Groups and Fundamental Groups"? That's usually a good place to look for things like this. $\endgroup$ – Alexander Betts Jul 19 '20 at 7:45
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