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

When the characteristic of $K$ does not divide $m\gt 1$, we may identify the finite abelian group $E[m]$ with $(\Z/m\Z)^2$ and hence view the representation as a map
\[
\overline\rho_{E,m}: \Gal(\overline{K}/K) \to \GL(2,\Z/m\Z)
\]
defined up to conjugation. In particular, when $m=\ell$ is a prime different from the characteristic of $K$, we have the **mod-$\ell$ Galois representation**
\[
\overline\rho_{E,\ell}: \Gal(\overline{K}/K) \to \GL(2,\Z/\ell\Z).
\]
Taking the inverse limit over prime powers $m=\ell^n$ yields the **$\ell$-adic Galois representation** attached to $E$,
\[
\rho_{E,\ell}: \Gal(\overline{K}/K) \to \Aut(T_\ell(E)) \cong \GL(2,\Z_\ell),
\]
which describes the action of the absolute Galois group of $K$ on $T_\ell(E)$, the $\ell$-adic Tate module of $E$.

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**
\[
\rho_{E}: \Gal(\overline{K}/K) \to \GL(2,\hat \Z).
\]
If $E$ is an elliptic curve without complex multiplication that is defined over a number field, then the image of $\rho_E$ is an open subgroup of $\GL(2,\hat\Z)$ that has an associated level, index, and genus.

**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2023-07-08 18:26:46

**Referred to by:**

- columns.ec_curvedata.modell_images
- ec.galois_rep_adelic_image
- ec.galois_rep_elladic_image
- ec.galois_rep_modell_image
- ec.maximal_elladic_galois_rep
- ec.maximal_galois_rep
- ec.q.1944.e1.bottom
- g2c.galois_rep_image
- modcurve.level_structure
- rcs.rigor.ec.q
- lmfdb/ecnf/templates/ecnf-curve.html (line 409)
- lmfdb/ecnf/templates/ecnf-curve.html (line 421)
- lmfdb/elliptic_curves/elliptic_curve.py (lines 451-453)
- lmfdb/elliptic_curves/elliptic_curve.py (line 467)
- lmfdb/elliptic_curves/templates/ec-curve.html (lines 430-433)
- lmfdb/elliptic_curves/templates/ec-curve.html (line 463)

**History:**(expand/hide all)

- 2023-07-08 18:26:46 by Andrew Sutherland (Reviewed)
- 2022-02-09 19:48:52 by Andrew Sutherland (Reviewed)
- 2022-02-02 19:49:36 by Shiva Chidambaram (Reviewed)
- 2021-09-18 10:18:51 by Andrew Sutherland (Reviewed)
- 2021-07-19 14:09:58 by Andrew Sutherland (Reviewed)
- 2021-07-17 12:51:14 by Andrew Sutherland
- 2018-06-19 22:22:36 by John Jones (Reviewed)

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