REMARKS ON A CONJECTURE OF FONTAINE AND MAZUR
We show that a continuous, odd, regular (non-exceptional), ordinary, irreducible, two-dimensional, $l$-adic representation of the absolute Galois group of the rational numbers is modular over some totally real field. We deduce that it occurs in the $l$-adic cohomology of some variety over the rationals and that its $L$-function has meromorphic continuation to the whole complex plane and satisfies the expected functional equation.
AMS 2000 Mathematics subject classification: Primary 11F80. Secondary 11G40(Received May 1 2001)
(Accepted May 30 2001)
Key Words: Galois representation; modularity; Fontaine–Mazur.