This is the exercise on Serre's book "l-adic abelian representations". on Section I-5. Notation: Galois group $G$ acts on $T_{\ell}(E)$, the Tate module representation, $G_{\ell}$ is the image of $G$ in $GL(T_{\ell}(E))$of the represention.

- There exists e.c. defined over Q(T) with j-invariant T.
- for the above curve, consider it to be defined over $C(T)$, the the image of $Gal_{C(T)}$ is $SL(T_{\ell}(E))$.
- over Q(T), the image of $Gal_{Q(T)}$ is $GL(T_{\ell}(E))$.
- for any closed subgroup H of $GL(T_{\ell}(E))$, there exists some e.c. defined over some field, with $G_{\ell}=H$.

The first one is easy. but I have no idea about the 2nd and 3rd one.