Example 2.2. Let E/Q be the elliptic curve defined by the Weierstrass equa-
+ y =
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) =

where q =
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)
: gcd(n, N) = 1}). The second
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