Companion forms and weight one forms. (English) Zbl 0965.11019

Buzzard, Kevin; Taylor, Richard

Ann. Math. (2) 149, No. 3, 905-919 (1999).

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 \(G_{\mathbb Q}\) of \(\mathbb Q\) [see Pac. J. Math. 1997, Spec. Issue, 337-347 (1997; Zbl 0942.11031)].
The main theorem of the paper proves that a continuous 2-dimensional \(\ell\)-adic representation \(\rho\) of \(G_{\mathbb Q}\) (\(\ell > 3\)) that is unramified at almost all primes, that is residually modular and absolutely irreducible, and that at \(\ell\) is unramified with eigenvalues that are residually distinct arises from a holomorphic weight one newform. That \(\rho\) arises from an overconvergent form of weight one follows without much difficulty from the results of Wiles et al. [see B. Mazur and A. Wiles, Compos. Math. 59, 231-264 (1986; 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”.

Reviewer: Chandrashekhar B.Khare (Mumbai)

MSC:

11F33	Congruences for modular and \(p\)-adic modular forms
11F11	Holomorphic modular forms of integral weight
11F80	Galois representations

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

Citations:

Zbl 0942.11031; Zbl 0654.12008