Could someone please recommend a good introductory text on Galois representations? In particular, something that might help with reading Serre's "Abelian l-Adic Representations and Elliptic Curves" and "Propriétés galoisiennes des points d'ordre fini des courbes elliptiques".
[Math] Introductory text on Galois representations
galois-representationsnt.number-theoryreference-requesttextbook-recommendation
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If you want to go further in understanding this point of view, I would advise you to begin learning class field theory. It is a deep subject, it can be understood in a vast variety of ways, from the very concrete and elementary to the very abstract, and although superficially it appears to be limited to describing abelian reciprocity laws, it in fact plays a crucial role in the study of non-abelian reciprocity laws as well.
The texts:
Ireland and Rosen for basic algebraic number theory, a Galois-theoretic proof of quadratic reciprocity, and other assorted attractions.
Cox's book on primes of the form x^2 + n y^2 for an indication of what some of the content of class field theory is in elementary terms, via many wonderful examples.
Serre's Local Fields for learning the Galois theory of local fields
Cassels and Frolich for learning global class field theory
The standard book at the graduate level to learn the arithmetic of non-abelian (at least 2-dimensional) reciprocity laws is Modular forms and Fermat's Last Theorem, a textbook on the proof by Taylor and Wiles of FLT. But it is at a higher level again.
I don't think that you will find a single text on this topic at a basic level (if basic means Course in arithmetic or Ireland and Rosen), because there is not much to say beyond what you stated in your question without getting into the theory of elliptic curves and/or the theory of modular forms and/or a serious discussion of class field theory.
Also, as basic suggests, you could talk to the grad students in your town, if not at your institution, then at the other one down the Charles river, which as you probably know is currently the world centre for research on non-abelian reciprocity laws (maybe shared with Paris). Certainly there are grad courses offered on this topic there on a regular basis.
There is a map from the $\mathbb P^1$ with coordinate $\gamma_2$ to the $\mathbb P^1$ with coordinate $j$ given by $j= \gamma_2^3$. We want to check that the fiber over $j$ has a $\mathbb Q(j)$ rational point. Because this is cubic covering, it can't gain a rational point over a quadratic extension if it didn't have one already, so for simplicity we can check that the fiber has a $\mathbb Q( j, \tau , \sqrt{-3})$-rational point.
Over $\mathbb Q(\mu_3)$, the $3$-torsion points define an $SL_2(\mathbb Z/3)$-covering of the modular curve, and this map is simply the map associated by the Galois correspondence to the subgroup $\Gamma(\gamma_2)$ of $SL_2(\mathbb Z/3)$. So by the Galois correspondence, the fiber has a rational point if and only if the action of the absolute Galois group of $\mathbb Q(j, \tau, \sqrt{-3})$ on the fiber factors through $\Gamma(\gamma_2)$.
To check this, we split into two cases depending on whether $3$ is split or inert in $\tau$. In either case, there is a natural action of the ring of integers of $\mathbb Q(\tau)$ on the $3$-torsion points which the Galois group commutes with. The action necessarily factors through the ring of integers mod $3$. If $3$ is split, the ring of integers mod $3$ is $\mathbb F_3 \times \mathbb F_3$, acting diagonalizably, and so the Galois group consists of elements that commute with it, which must lie in $\mathbb F_3^\times \times \mathbb F_3^\times$ (or the determinant $1$ elements of it). If $3$ is inert, the ring is $\mathbb F_9$, and so the Galois group must lie in $\mathbb F_9^\times$ (or the determinant $1$ elements of it). No matter how one of these two subgroups is conjugated, they must lie in the subgroup you have written down (for instance because it contains all the elements of order a power of $2$).
The group $\mathbb F_3^\times \times \mathbb F_3^\times$ here is a split Cartan, as are all its conjugates, and $\mathbb F_9^\times$ and all its conjugates are non-split Cartans.
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Kevin Ventullo's suggestion of Silverman's book is a very good one. The first examples of Galois representations in nature are Tate modules of elliptic curves, and if you haven't read about them in Silverman's book, you should.
If you have read Silverman's book, a nice paper to read is Serre and Tate's "On the good reduction of abelian varieties". It is a research paper, not a text-book, and is at a higher level than Silverman (especially in its use of algebraic geometry), but it has the merit of being short and beautifully written, and uses Galois representation techniques throughout.
One fantastic paper is Swinnerton-Dyer's article in Lecture Notes 350. Here he explains various things about the Galois representations attached to modular forms. The existence of the Galois representations is taken as a black box, but he explains the Galois theoretic significance of various congruences on the coefficients of the modular forms. Reading it is a good way to get a concrete feeling of what Galois representations are and how you can think about and argue with them.
Another source is Ken Ribet's article "Galois representations attached to modular forms with nebentypus" (or something like that) in one of the later Antwerp volumes. It presupposes some understanding of modular forms, but this would be wise to obtain anyway if you want to learn about elliptic curves, and again demonstrates lots of Galois representation techniques. It would be a good sequel to Swinnerton-Dyer's article.
Yet another good article to read is Ribet's "Converse to Herbrand's criterion" article, which is a real classic. It is reasonably accessible if you know class field theory, know a little bit about Jacobians (or are willing to take some results on faith, using your knowledge of elliptic curves as an intuitive guide), and something about modular forms. Mazur recently wrote a very nice article surveying Ribet's, available here on his web-site.
One problem with reading Serre is that he uses $p$-adic Hodge theory in a strong way, but his language is a bit old-fashioned and out-dated (he was writing at a time when this theory was in its infancy); what he calls "locally algebraic" representations would now be called Hodge--Tate representations. To learn the modern formulation of and perspective on $p$-adic Hodge theory you can look at Laurent Berger's various exposes, available on his web-site. (This will tell you much more than you need to know for Serre's book, though.)
For a two page introduction to Galois representation theory, you could read Mark Kisin's What is ... a Galois representation? for a two-page introduction.
Yet another source is the Fermat's Last Theorem book (Cornell--Silverman--Stevens), which has many articles related to Galois representations, some more accessible than others.
The article of Taylor that Chandan mentioned in a comment is also very nice, although it moves at a fairly rapid clip if you haven't seen any of it before.
Serre's article in Duke 54, in which he explains his conjecture about the modularity of 2-dimensional mod p Galois representations, is also very beautiful, and involves various concrete computations which could be helpful
One last remark: if you do want to understand Galois representations, you will need to have a good understanding of the structure of the Galois groups of local fields (as described e.g. in Serre's book "Local fields"), in particular the role of the Frobenius element, of the inertia subgroup, and of the significance of tame and wild inertia.