Question

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.

1. There exists e.c. defined over Q(T) with j-invariant T.
2. for the above curve, consider it to be defined over $C(T)$, the the image of $Gal_{C(T)}$ is $SL(T_{\ell}(E))$.
3. over Q(T), the image of $Gal_{Q(T)}$ is $GL(T_{\ell}(E))$.
4. 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.

Answer 1

The idea for (2) is the following: the modular curve $Y(\ell^n)$ classifying elliptic curves over ${\mathbb C}$ together with an isomorphism $({\mathbb Z}/\ell^n)^2 \cong E[\ell^n]$ identifying the standard symplectic pairing on the left (i.e. $\langle (a_1,a_2),(b_1 ,b_2)\rangle = e^{2\pi i (a_1b_2-a_2b_1)/\ell^n}$) with the Weil pairing on the right, is irreducible. (It is isomorphic to $\mathcal H/\Gamma(\ell^n)$, where $\mathcal H$ is the complex upper half-plane and $\Gamma(\ell^n)$ is the congruence subgroup of full level $\ell^n$.)

(3) follows from (2) and the irreducibility of cyclotomic polynomials over ${\mathbb Q}$.

Answer 2

I exceedingly concur with Emerton on the nontriviality of this problem. The ideas for its solution take up the bulk of chapter 7 in Diamond and Shurman's book on modular forms.