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A Galois representation associated to an elliptic curve $E$ over a number field $K$ is said to be maximal if its image is as large as possible subject to the constraints imposed by the field $K$ and the endomorphism ring of $E$.

The constraint imposed by $K$ is determined by its intersection with cyclotomic extensions of $\Q$. In particular, if $p$ is a prime then the representation \[ \bar\rho_{E,p}\colon \Gal(\overline{K}/K)\to\GL_2(\Z/p\Z)\overset{\det}{\rightarrow}GL_1(\Z/p\Z) \] factors through the faithful representation $\Gal(K(\zeta_p)/K)\to\GL_1(\Z/p\Z)$ induced by the cyclotomic character. The image in $\GL_1(\Z/p\Z)$ has cardinality $[K(\zeta_p):K]$, which is equal to $p-1$ when $p$ is unramified in $K$ (and in particular, when $K=\Q$).

If $E$ does not have complex multiplication then $\End(E)\simeq \Z$ does not impose any constraints and the image of $\bar\rho_{E,p}$ is maximal if and only if it contains $\SL_2(\Z/p\Z)$.

If $E$ has complex multiplication then $\End(E)$ is isomorphic to an order $\mathcal{O}$ in an imaginary quadratic field $F$. This imposes constraints on the image of $\bar\rho_{E,p}$ that depend on whether $K$ contains $F$ or not, and on the quadratic character of the discriminant $D:=\mathrm{disc}(\mathcal{O}):=[\mathcal{O}_K:\mathcal{O}]^2\mathrm{disc}(F)$ modulo $p$.

If $p$ divides $D$ then $E$ admits a rational $p$-isogeny and the image of $\bar\rho_{E,p}$ must lie in a Borel subgroup of $\GL_2(\Z/p\Z)$ and is maximal if and only if it contains a Borel subgroup of $\SL_2(\Z/p\Z)$.

If $p$ does not divide $D$ and $K$ contains $F$ then the image of $\bar\rho_{E,p}$ is abelian and must lie in a Cartan subgroup $C$ of $\GL_2(\Z/p\Z)$. When $\left(\frac{D}{p}\right)=+1$ this is a split Cartan subgroup, and when $\left(\frac{D}{p}\right)=-1$ it is a non-split Cartan subgroup. The image of $\rho_{E,p}$ is maximal if and only if it contains the intersection of $C$ and $\SL_2(\Z/p\Z)$.

If $p$ does not divide $D$ and $K$ does not contain $F$ then the image of $\bar\rho_{E,p}$ lies in a 2-extension $N$ of a Cartan subgroup $C$ which is split or non-split depending on $\left(\frac{D}{p}\right)=\pm 1$ as above. If $p$ is odd then $N$ is the normalizer of $C$ in $\GL_2(\Z/p\Z)$, and for $p=2$ the group $N$ is determined up to conjugacy by its cardinality (2 or 6). The image of $\rho_{E,p}$ is maximal if and only if it contains the intersection of $N$ and $\SL_2(\Z/p\Z)$.

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  • Review status: reviewed
  • Last edited by John Jones on 2018-06-19 20:11:23
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