Let be an elliptic curve over , without complex
multiplication over
. For a prime number,
consider the representation
induced by the Galois action on
the group of -torsion points of . A theorem of Serre, published
in 1972, asserts that there exists an integer such that the
above representation is surjective for larger than . Serre
then asked the following question: can be chosen independently
of ? The classification of maximal subgroups of
shows that this boils down to proving the triviality, for large enough
, of the sets of rational points of four families of modular
curves, namely ,
,
and
(we say that a
point of one of those curves is *trivial* if it is either a cusp,
or the underlying isomorphism class of elliptic curves has complex
multiplication over
). The (so-called exceptional)
case of
was ruled out by Serre. The fact that
is made of only cusps for is a well-known
theorem of Mazur. In this talk we will present a proof that
is trivial for large enough (joint
work with Yuri Bilu).