Potential Modularity and Applications

In our seminar we have been working towards a modularity lifting theorem. Recall that such a theorem allows one (under suitable hypotheses) to deduce the modularity of a p-adic Galois representation from that of the corresponding mod p representation. This is a wonderful theorem, but it is not immediately apparent how it can be applied: when does one know that the residual representation is modular? One example where residual modularity is known is the following: a theorem of Langlands and Tunnel states that any Galois representation (of any number field) into GL2(F3) is modular. Their result is specific to F3 and does not apply to representations valued in other finite fields (except perhaps F2?): the key point is that GL2(F3) is solvable. Modularity lifting thus allows one to conclude (under appropriate hypotheses) that representations into GL2(Z3) are modular. Wiles’ original application of modularity lifting to elliptic curves used this line of reasoning. For finite fields other than F3 (and maybe F2) there is no analogue of the Langlands–Tunnel theorem: the finite groups GL2(Fq) are typically not solvable. However, Taylor [Tay], [Tay2] partially found a way around this problem: he observed, using a result of Moret-Bailly, that any odd residual representation of a totally real field F becomes modular after passing to a finite extension of F ; that is, odd residual representations of F are potentially modular. Using modularity lifting, one can conclude that many p-adic are potentially modular as well. Typically, one cannot deduce modularity from potential modularity. Nonetheless, many of the nice properties of modular p-adic representations can be established for potentially modular representations as well: they satisfy the Weil bounds, their L-functions admit meromorphic continuation and satisfy a functional equation, they often can be realized in the Tate module of an abelian variety and they fit into compatible systems. We prove the final of these results. As if these consequences of potential modularity were not impressive enough, Khare and Wintenberger [KW] went even farther: they proved that every irreducible odd residual representation of GQ is modular, a result first conjectured by Serre. To do this, they first showed — using potential modularity — that any mod p representation admits a nice p-adic lift. This lift (by one of the corollaries of potential modularity) fits into a compatible system. To prove the modularity of the original mod p representation, it suffices (by modularity lifting, and basic properties of compatible systems) to prove the modularity of the reduction of any of the `-adic representations in the system. This permits the possibility of an inductive argument, which turns out to be quite subtle but possible. The base cases of the induction had been previously proved by Serre and Tate; these results are specific to Q and is one reason that this sort of result has not been extended to other fields.