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

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”.

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 |