One classical example is the group of Q-defined -torsion points of an elliptic curve
E defined over Q. We will treat this example in Section 2. In this survey we
will be interested in (compatible systems of) Galois representations arising from
automorphic representations. In Section 4 we will describe Galois representations
attached to an automorphic representation π which satisfies several technical con-
ditions. The statements of the most recent results (to the best of my knowledge)
on the inverse Galois problem obtained by means of compatible systems of Galois
representations attached to automorphic representations can be found in Section 5,
together with some ideas about their proofs.
A remarkable feature of this method is that, in addition, one obtains some
control of the ramification of the Galois extension that is produced. Namely, it will
only be ramified at the residual characteristic and at a finite set of auxiliary primes,
that usually one is allowed to choose (inside some positive density set of primes).
This will be highlighted in the statements below.
2. Some classical cases
In this section we revisit some classical examples of Galois representations at-
tached to geometric objects. We begin with the Galois representations attached to
the torsion points of elliptic curves, and later we will see them as a particular case
of Galois representations attached to modular forms.
2.1. Elliptic curves. An elliptic curve is a genus one curve, endowed with a
distinguished base point. Every elliptic curve E can be described by means of a
Weierstrass equation, that is, an affine equation of the form
+ a1xy + a3y =
+ a4x + a6
where the coefficients a1,...,a6 lie in some field K. The most significant property of
elliptic curves is that the set of points of E (defined over some field extension L/K)
can be endowed with a commutative group structure, where the neutral element is
the distinguished base point.
Let E/Q be an elliptic curve and let be a prime number. We can consider the
subgroup E[ ](Q) of E(Q) consisting of -torsion points. This group is isomorphic
to the product of two copies of F . Moreover, since the elliptic curve is defined over
Q, the absolute Galois group GQ acts naturally on the set of Q-defined points of
E, and this action restricts to E[ ](Q). We obtain thus a Galois representation
ρE, : GQ Aut(E[ ](Q)) GL2(F ).
As explained in the introduction, the image of ρE, can be realised as a Galois
group over Q. This brings forward the question of determining the image of such a
Galois representation. In this context, there is a classical result by J. P. Serre from
the seventies ([29], Th´ eor` eme 2).
Theorem 2.1 (Serre). Let E/Q be an elliptic curve without complex multi-
plication over Q. Then the representation ρE, is surjective for all except finitely
many primes .
We can immediately conclude that GL2(F ) can be realised as a Galois group
over Q for all except finitely many primes . However, we can do even better by
picking a particular elliptic curve and analysing the Galois representations attached
to it.
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