Companion forms and weight one forms

Kevin Buzzard and Richard Taylor

Review from Zentralblatt MATH:

This paper is an important link in the second author's programme that has been successful in proving very many cases of Artin's conjecture on the holomorphy of $L$-series attached to non-trivial irreducible 2-dimensional complex representations of the Galois group $G_{\Bbb Q}$ of $\Bbb Q$ [see Pac. J. Math. 1997, Spec. Issue, 337-347 (1997; Zbl 0654.12008)].

The main contribution of this paper is a beautiful argument that proves that under the above hypotheses the overconvergent form is indeed a classical (holomorphic) form of weight one. The authors prove this by studying the rigid analytic geometry of modular curves, and invoking ``rigid GAGA''.

Reviewed by Chandrashekhar B.Khare

Keywords: Artin's conjecture; Galois group; $\ell$-adic representation; holomorphic weight one newform; overconvergent form of weight one; rigid analytic geometry of modular curves; rigid GAGA

Classification (MSC2000): 11F33 11F11 11F80

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