Base change, lifting, and Serre's conjecture (Q1355099)

Let \(F\) be a number field, \(p\) a rational prime, \(S\) a finite set of places of \(F\) including those above \(p\) and the archimedean places, \(F_S\) the maximal algebraic extension of \(F\) in \(\mathbb C\) which is unramified outside \(S\). Suppose that \(\rho:\text{Gal}(F_S/F)\to GL(2,{\overline {\mathbb F}}_p)\) is an irreducible continuous representation with values in \(GL(2,\mathbb F)\), where \(\mathbb F\) is a finite extension of \(\mathbb F_p\). The main result asserts that there is a finite set \(S'\) containing \(S\) of places of \(F\) and a representation \(\rho':\text{Gal}(F_{S'}/F)\to GL(2,R/p^2)\), where \(R\) is the Witt ring of \(\mathbb F\), which reduces to \(\rho\) mod \(p\). It is noted that if \(\rho\) is reducible, it always lifts to \(\rho':\text{Gal}(F_{S'}/F)\to GL(2,R)\). This is motivated by Serre's conjecture that an odd continuous irreducible representation \(\rho:\text{Gal}({\overline{\mathbb Q}}/\mathbb Q)\to GL(2,{\overline{\mathbb F}}_p)\) is modular, which implies that such \(\rho:\text{Gal}(\mathbb {Q}_S/\mathbb Q)\to GL(2,{\overline{\mathbb F}}_p)\) lifts to \(\rho':\text{Gal}({\mathbb Q}_S/\mathbb Q) \to GL(2, R_K)\), where \(R_K\) is the ring of integers in a finite extension \(K\) of \({\mathbb Q}_p\). The technique of this note would not extend beyond mod \(p^2\). Reviewer's remark: To improve the exposition, the note -- instead of recalling on the 3rd page the notations of the 1st page -- could recall the definition of the Witt ring, refer to other works in the area (e.g. that mentioned at the bottom of the 1st page), clarify what refinements are meant on the 1st page, arrange the claim of the Lemma on the 2nd page in a logical order (and provide a reference for the proof), remove the witticism in the middle of the last page, and perhaps speculate less vaguely -- or not at all -- also in the context of \(GL(n)\).