The Fontaine-Mazur conjecture predicts that a two dimensional p-adic Galois representation of the absolute Galois group of Q which is irreducible, unramified outside finitely many places, and potentially semi-stable at p, comes from a modular form.
This is a remarkable conjecture, because it says that a local condition at p implies that the eigenvalues of Frobenius elements are Weil numbers,
and that there is well defined and entire complex L-function attached to the Galois representation.
I will explain a result which confirms this conjecture for representations coming from finite slope overconvergent modular forms: If such a representation is potentially semi-stable then (upto a twist) it comes from a classical form.