On lifting and modularity of reducible residual Galois representations over imaginary quadratic fields

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 our previous work we do not impose any restrictions on the dimension of the crystalline Selmer group $H_{\Sigma }^1(F,\mathrm{Hom}({\tau }_2,{\tau }_1))\subset {\mathrm{Ext}}^1({\tau }_2,{\tau }_1)$. We establish that there exists a basis $\mathcal{B}$ of $H_{\Sigma }^1(F,\mathrm{Hom}({\tau }_2,{\tau }_1))$ 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).

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

References in corpus (1)

• p-adic families of Galois representations and higher rank Selmer groups

Cited by in corpus (1)

• The Eisenstein ideal with squarefree level