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.)
1 Answer
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.
