Let E be an elliptic curve over the rationals, and let $TE = \lim\_\leftarrow E[n]$ be the Tate module of the elliptic curve. The action of the Galois group of $\bf Q$ gives rise to a representation $\rho\_E : G\_{\bf Q} \rightarrow GL\_2(\widehat{\bf Z})$. 
My first question is why is the image of this representation not surjective? I seem to recall there is a very easy argument for it, but I can't remember what that is.
My second question is can one use this to give a level structure to all rational elliptic curves? Specifically, can I find a collection of modular curves with some level structure $\{X\_i/{\bf Q}\}$ such that $\cup j(X\_i({\bf Q})) = X(1)({\bf Q})$, where $j:X\_i \rightarrow X(1)$ is the natural forgetful map? (I'm sure the answer to this is no, but it never hurts to be overly optimistic.)



Images of action of Galois on the Tate module of Elliptic Curve,
Let $\Delta$ be the discriminant of $E$. Then the action of $G\_{\mathbf{Q}}$ on $E[2]$ determines the action on $\sqrt{\Delta}$. On the other hand, the action of $G\_{\mathbf{Q}}$ on $E[n]$ determines the action on a primitive $n$-th root of unity $\zeta\_n$, via the Weil pairing. The Kronecker-Weber theorem implies that $\sqrt{\Delta} \in \mathbf{Q}(\zeta\_n)$ for some $n$, and this forces a compatibility between the actions on $E[2]$ and $E[n]$, which forces the image of $G\_{\mathbf{Q}}$ into an index-2 subgroup of $\operatorname{GL}\_2(\hat{\mathbf{Z}})$. But this index-2 subgroup varies with $E$, so you do not get a rational point on a nontrivial modular curve corresponding to any *one* kind of level structure.


**Remarks:** 


1) Nathan Jones in his Ph.D. thesis at UCLA, building on earlier work of William Duke, proved that in a precise sense, asymptotically `100%` of elliptic curves are such that the image of Galois is of index 2. 


2) Over higher number fields $K$, quadratic extensions are not necessarily contained in cyclotomic ones, and in fact there exist elliptic curves over certain number fields $K$ other than $\mathbf{Q}$ for which $G\_K \to \operatorname{GL}\_2(\hat{\mathbf{Z}})$ is surjective. The first example was given by Aaron Greicius in his Ph.D. thesis. 


3) David Zywina then proved that under mild necessary conditions on a number field $K$, asymptotically `100%` of elliptic curves are such that $G\_K \to \operatorname{GL}\_2(\hat{\mathbf{Z}})$ is surjective.


