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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.

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MO is not the right place to ask questions along the lines of "how do i do this exercise?", really. You can turn it into a good question, maybe, by explaining what you tried, and so on. Read the link on the top of the page labeled 'how to ask' for more tips. –  Mariano Suárez-Alvarez Jan 29 '10 at 7:42
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This is not a trivial homework exercise. –  Emerton Jan 29 '10 at 13:05
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Mariano, I have to disagree with you here. Exercises like this may well be of interest to mathematicians (I know little number theory, so can't comment specifically) and we've had questions on problems out of Atiyah-Macdonald from people who needed them for other things and from self-learners before. Yes, basic could have given more background, but exercises in advanced books are certainly within the purview of Math Overflow. –  Charles Siegel Jan 29 '10 at 13:10
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Serre's book is not a textbook. The ideas for the solution are not contained in the book. –  Emerton Jan 29 '10 at 17:05
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Yes, but as I wrote, Serre's book is not a text-book, it is a research monograph. The questioner is asking ``why is this statement in this research monograph true''? –  Emerton Jan 29 '10 at 18:22
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2 Answers 2

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 ${\mathbb Q}$.

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I started reading Serre's book, got stumped on this question, and now am in awe after reading these answers! The results in chapter 7 of Diamond and Shurman's book are currently a bit over my head. Some even seem much newer than Serre's book. So, this comment is to ask for your advice - should I read A First Course before Abelian l-adic Representations? –  Dror Speiser May 21 '10 at 15:56
    
I don't know the contents of Diamond--Shurman, so can't really comment. You could also try Shimura's book, or some of Lang's books (he wrote several that involve elliptic curves in various ways). –  Emerton May 21 '10 at 18:48
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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.

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+1 stankewicz: you should know. (This will be the last time I give an upvote to someone mostly because s/he is my PhD student.) –  Pete L. Clark Jan 29 '10 at 19:10
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