# $\ell$-adic Representations of Elliptic Curves

If you’re somehow seeing this right now, look away. It’s not finished, and I’m not sure when/if it will be

The title of this post is technically incorrect. We won’t be talking about representations of elliptic curves, but about representations attached to (associated with?) elliptic curves. In particular, the goal of this post is to prove that given an elliptic curve $E/\Q$ defined over the rationals, and a prime $\l$, the $\l$-adic representation $G_{\Q}\to\GL(T_\l(E))\iso\GL_2(\Z_\l)\subset\GL_2(\Q_\l)$ is irreducible where $G_{\Q}=\Gal(\Qbar/\Q)$ is the absolute Galois group of $\Q$. If you don’t know what some of these words mean, don’t worry; I’ll explain.