# Outer Galois representations and Tate modules of Jacobian varieties

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.

• 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. Jul 19, 2020 at 7:45