4 SARA ARIAS-DE-REYNA Example 2.2. Let E/Q be the elliptic curve defined by the Weierstrass equa- tion y2 + y = x3 x. This curve is labelled 37A in [6], and it has the property that ρE, is surjective for all primes (see [29], Example 5.5.6). Therefore we obtain that GL2(F ) occurs as the Galois group of a finite Galois extension K/Q. Moreover, we have additional information on the ramification of K/Q namely, it ramifies only at 37 (which is the conductor of E) and . The next situation we want to analyse is that of Galois representations attached to modular forms. Let us recall that modular forms are holomorphic functions defined on the complex upper half plane, which satisfy certain symmetry relations. We will not recall here the details of the definition (see e.g. [9] for a complete treatment focusing on the relationship with arithmetic geometry). These objects, of complex-analytic nature, play a central role in number theory. At the core of this relationship is the fact that one can attach Galois representations of GQ to them. More precisely, let f be a cuspidal modular form of weight k 2, conductor N and character ψ (in short: f Sk(N, ψ)), which is a normalised Hecke eigenform. We may write the Fourier expansion of f as f(z) = n≥1 anqn, where q = e2πiz. A first remark is that the coefficient field Qf = Q({an : gcd(n, N) = 1}) is a number field. Denote by OQf its ring of integers. By a result of Deligne (cf. [7]), for each prime λ of OQf there exists a (continuous) Galois representation ρf,λ : GQ GL2(OQf,λ ), related to f, where Qf,λ denotes the completion of Qf at the prime λ, Qf,λ an algebraic closure thereof and OQf,λ is the valuation ring of Qf,λ. Here the topology considered on GL2(OQf,λ ) is the one induced by the -adic valuation. The relationship between ρf,λ and f is the following. First, ρf,λ is unramified outside N. Moreover, for each p N, we can consider the image under ρf,λ of a lift Frobp of a Frobenius element at p (this is well defined because ρf,λ is unramified at p). Then the characteristic polynomial of ρf,λ(Frobp) equals T 2 apT + ψ(p)pk−1. We may compose each ρf,λ with the reduction modulo the maximal ideal of OQf,λ , and we obtain a (residual) representation ρf,λ : GQ GL2(κ(Qf,λ)) GL2(F ), where is the rational prime below λ. One of the main recent achievements in number theory has been the proof of Serre’s Modularity Conjecture, which says that every Galois representation ρ : GQ GL2(F ) which is odd and irreducible is actually isomorphic to ρf,λ for some modular form f and some prime λ as above. In this survey we are interested in the image of ρf,λ. These images have been studied by K. Ribet (cf. [24], [25]). One first remark is that, when ρf,λ is absolutely irreducible, then ρf,λ can be conjugated (inside GL2(OQf,λ )) so that its image is contained in GL2(OQf,λ ). Therefore, in this case we can assume that ρf,λ : GQ GL2(κ(Qf,λ)), where κ(Qf,λ) denotes the residue field of Qf,λ. To state Ribet’s result, we first introduce two more number fields related to f. The first one is the twist invariant coefficient field of f, which is the subfield of the coefficient field of f defined as Ff := Q({an/ψ(n) 2 : gcd(n, N) = 1}). The second
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