On Lifting and Modularity of Reducible Residual Galois Representations Over Imaginary Quadratic Fields. (18th January 2015)

• Record Type: Journal Article
• Authors: Berger, Tobias; Klosin, Krzysztof
• Is Part Of: International mathematics research notices. Volume 2015:Number 20(2015)
• Page Start: 10525
• Page End: 10562
• Publication Date: 2015-01-18
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnu266
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract : In this paper, we study deformations of mod $p$ Galois representations $\tau$ (over an imaginary quadratic field $F$ ) of dimension $2$ whose semi-simplification is the direct sum of two characters $\tau _1$ and $\tau _2$ . As opposed to [7 ], we do not impose any restrictions on the dimension of the crystalline Selmer group $H^1_{\Sigma }(F, {\rm Hom}(\tau _2, \tau _1)) \subset {\rm Ext}^1(\tau _2, \tau _1)$ . We establish that there exists a basis ${\mathcal {B}}$ of $H^1_{\Sigma }(F, {\rm Hom}(\tau _2, \tau _1))$ arising from automorphic representations over $F$ (Theorem 8.1). Assuming among other things that the elements of ${\mathcal {B}}$ admit only finitely many crystalline characteristic 0 deformations, we prove a modularity lifting theorem asserting that if $\tau$ itself is modular then so is its every crystalline characteristic zero deformation (Theorems 8.2 and 8.5).