Journal of the Institute of Mathematics of Jussieu



REMARKS ON A CONJECTURE OF FONTAINE AND MAZUR


Richard Taylor a1
a1 Department of Mathematics, Harvard University, Cambridge, MA 02138, USA

Abstract

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.