irreduciblity of $\ell$-adic representation attach to the elliptic curve over $\mathbf{Q}$ with complex multiplication

Question

Currently I am reading the book, Fermat’s Last Theorem written by Darmon, Diamond and Taylor. (You can find this pdf online http://www.math.mcgill.ca/darmon/pub/Articles/Expository/05.DDT/paper.pdf) In proposition 2.8 part (b), it says if one has an elliptic curve defined over $\mathbf{Q}$, for a prime number $\ell$, consider the $\ell$-adic representation attached to the tate module of this elliptic curve: $$\rho_{E,l}:G_{\mathbf{Q}}\rightarrow GL_{2}(\mathbf{Z}_{\ell})$$ Then $\rho_{E,\ell}$ is absolutely irreducible for all $\ell$. The proof of this statement refers to Serre's book 'Abelian l-adic Representations and Elliptic Curves'. However, in that book, it only shows the irreducibility for an elliptic curve without complex multiplication. I am wondering the idea of proving the irreducibility in complex multiplication case; any references would be appreciated.

Answer 1

An elliptic curve over $\mathbf Q$ cannot have complex multiplication (defined over $\mathbf Q$). It's possible for a rational elliptic curve to have extra endomorphisms, but these will only be defined over a finite extension.

But let's instead take an elliptic curve $E$ over a number field $K$ with complex multiplication. Then the associated Galois representation is reducible*!

Indeed, if $$\rho_{E,\ell}:G_K\to \mathrm{GL}_2(\overline{\mathbf Q}_\ell)$$ is the associated $\ell$-adic representation, then its not too hard to check that $$\mathrm{End}(E)\otimes_\mathbf Z\overline{\mathbf Q}_\ell\hookrightarrow\mathrm{End}(\rho_{E,\ell}).$$

In particular, if $\mathrm{End}(E)\ne \mathbf Z$, then $\mathrm{End}(\rho_{E,\ell})$ is not a field, so $\rho_{E,\ell}$ is reducible. Its subrepresentations are one dimensional Galois representations, which by class field theory, correspond to the Grossencharacters of $E$.

In fact the above map is an isomorphism (by Faltings' theorem). So if $\mathrm{End}(E) = \mathbf Z$, then $\mathrm{End}(\rho_{E,\ell})$ is a field, so $\rho_{E,\ell}$ is irreducible.

If $E$ does not have complex multiplcation over $K$, but obtains extra endomorphisms over a finite extension, then the above argument shows that $\rho_{E, \ell}$ is irreducible. However, $\rho_{E, \ell}$ will not be surjective. By Mackey theory, since $\rho_{E, \ell}$ is irreducible, but $\rho_{E, \ell}|_{G_L}$ is reducible for some $L$, we find that $\rho_{E, \ell}$ is induced from a character of a quadratic extension. In particular, its image cannot be $\mathrm{GL}_2(\mathbf Z_\ell)$.

*By reducible, I mean that it becomes reducible over the algebraic closure $\overline{\mathbf Q}_\ell$. It may still be irreducible over $\mathbf Z_\ell$.