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