Galois representations attached to the Jacobian of a genus 2 curve (reviewed)

If $A$ is the Jacobian of a genus 2 curve $C$ defined over a field $K$ and $m$ is a positive integer, then the mod-$m$ Galois representation attached to $A$ is the continuous homomorphism \[ \overline\rho_{A,m}: \Gal(\overline{K}/K) \to \Aut(A[m]) \] describing the action of the absolute Galois group of $K$ on the $m$-torsion subgroup $A[m]$.

When the characteristic of $K$ does not divide $m\gt 1$, we may identify the finite abelian group $A[m]$ with $(\Z/m\Z)^4$. Since the Weil pairing is a non-degenerate, alternating, bilinear pairing $A[m] \times A[m] \to \mu_m$ equivariant with respect to the natural Galois action, we may view the representation as a map \[ \overline\rho_{A,m}: \Gal(\overline{K}/K) \to \GSp(4,\Z/m\Z) \] defined up to conjugation. In particular, when $m=\ell$ is prime different from the characteristic of $K$, we have the mod-$\ell$ Galois representation \[ \overline\rho_{A,\ell}: \Gal(\overline{K}/K) \to \GSp(4,\Z/\ell\Z). \] Taking the inverse limit over prime powers $m=\ell^n$ yields the $\ell$-adic Galois representation attached to $A$, \[ \rho_{A,\ell}: \Gal(\overline{K}/K) \to \Aut(T_\ell(E)) \cong \GSp(4,\Z_\ell), \] which describes the action of the absolute Galois group of $K$ on $T_\ell(A)$, the $\ell$-adic Tate module of $A$.

When $K$ has characteristic zero one can take the inverse limit over all positive integers $m$ (ordered by divisibility) to obtain the adelic Galois representation \[ \rho_{A}: \Gal(\overline{K}/K) \to \GSp(4,\hat \Z). \]