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Modular forms and Galois Representations

J. Tilouine (2002)

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

Modularity of an odd icosahedral representation

Arnaud Jehanne, Michael Müller (2000)

Journal de théorie des nombres de Bordeaux

In this paper, we prove that the representation $ρ$ from $G_{ℚ}$ in GL $_{2} (ℂ)$ with image $A_{5}$ in PGL $_{2} (A_{5})$ corresponding to the example $16$ in [B-K] is modular. This representation has conductor $5203$ and determinant $χ_{- 43}$ ; its modularity was not yet proved, since this representation does not satisfy the hypothesis of the theorems of [B-D-SB-T] and [Tay2].

Modularity of fibres in rigid local systems.

Darmon, Henri (1999)

Annals of Mathematics. Second Series

Modularity of Galois representations

Chris Skinner (2003)

Journal de théorie des nombres de Bordeaux

This paper is essentially the text of the author’s lecture at the 2001 Journées Arithmétiques. It addresses the problem of identifying in Galois-theoretic terms those two-dimensional, $p$ -adic Galois representations associated to holomorphic Hilbert modular newforms.

Modularity of $p$ -adic Galois representations via $p$ -adic approximations

Chandrashekhar Khare (2004)

Journal de Théorie des Nombres de Bordeaux

In this short note we give a new approach to proving modularity of $p$ -adic Galois representations using a method of $p$ -adic approximations. This recovers some of the well-known results of Wiles and Taylor in many, but not all, cases. A feature of the new approach is that it works directly with the $p$ -adic Galois representation whose modularity is sought to be established. The three main ingredients are a Galois cohomology technique of Ramakrishna, a level raising result due to Ribet, Diamond, Taylor,...

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