2015/01/01 by Tobias Berger, Krzysztof Klosin
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic number field #Automorphic form #Dimension (graph theory) #Field (mathematics) #Galois module #Homotopy and Cohomology in Algebraic Topology #Modular form #Modularity (biology) #Quadratic equation #Quadratic field #math.NT #msc:11F55 #msc:11F80
paper · pdf · doi:10.1093/imrn/rnu266
published as Int. Math. Res. Not. Volume 2015, Issue 20, 10525-10562 · 29 pages, this is a pre-copyedited, author-produced PDF of an article published in Int. Math. Res. Not. following peer review. The version of record is available online at: http://imrn.oxfordjournals.org/content/2015/20/10525
openalex publication_date 2015/01/01 · arxiv created 2016/06/21 · arxiv updated 2016/06/22 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
In this paper, we study deformations of mod |p| Galois representations |τ| (over an imaginary quadratic field |F|) of dimension |2| whose semi-simplification is the direct sum of two characters |τ 1| and |τ 2|. As opposed to [7], we do not impose any restrictions on the dimension of the crystalline Selmer group |H1Σ (F, \rm Hom(τ 2, τ 1)) ⊂ \rm Ext1(τ 2, τ 1)|. We establish that there exists a basis |\mathcal B| of |H1Σ (F, \rm Hom(τ 2, τ 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 |τ| itself is modular then so is its every crystalline characteristic zero deformation (Theorems 8.2 and 8.5).