Overconvergent Modular Forms and the Fontaine-Mazur conjecture
(Un exposé de Mark Kisin au séminaire de théorie des
nombres de Chevaleret le 28 janvier 2002)
Résumé :
The Fontaine-Mazur conjecture predicts that 2-dimensional p-adic
Galois
representations, whose restriction to a decomposition group at p satisfies
a certain
condition (potential semi-stability), come from modular forms.
I will explain a result which says that if a representation satisfies
this
condition
and comes from an overconvergent modular form, then it comes from a
classical modular form.
This is significant, because overconvergent modular forms are very closely
related
to classical ones. The fact that the potential semi-stability condition
can distinguish
between classical and non-classical forms, therefore provides a good
test
of the conjecture.