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 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**
\[
\rho_{A}: \Gal(\overline{K}/K) \to \GSp(4,\hat \Z).
\]

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**Knowl status:**

- Review status: beta
- Last edited by Shiva Chidambaram on 2022-02-02 19:44:38

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