We show that a continuous, odd, regular (non-exceptional), ordinary, irreducible, two-dimensional, -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 -function has meromorphic continuation to the whole complex plane and satisfies the expected functional equation.AMS 2000 Mathematics subject classification: Primary 11F80. Secondary 11G40
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Richard Taylor (2002) studied this question.