Let \(\rho: G_{\mathbf{Q},S} \rightarrow \mathrm{GL}_2(\overline{\mathbf{Q}}_p)\) be continuous and unramified at \(p\). The Fontaine-Mazur conjecture predicts that \(\rho\) has finite image and is automorphic. Buzzard and Taylor proved this result under the assumption the natural assumption that \(\rho\) is odd, that \(\overline{\rho}\) is modular (now uneccessary), but also under the futher two assumptions:
\(\overline{\rho}\) is irreducible,
\(\overline{\rho}\) is \(p\)-distinguished.
(caveat: there are mild extra assumptions required when \(p = 2\).) The point of this post is to note that is seems possible to remove either of these conditions and to wonder whether both can be removed simultaneously.
Suppose the second condition fails. One may enlarge the ordinary Hecke algebra \(\mathbf{T}\) to include the operator \(U_p\), call the resulting ring \(\widetilde{\mathbf{T}}\). After appropriate localizations, this is different from \(\mathbf{T}\) exactly when \(\overline{\rho}(\mathrm{Frob}_p)\) is a scalar. On the Galois side, let \(R_p\) denote the universal (framed) deformation ring. Then, for any lift of Frobenius, one can define the quadratic extension \(\widetilde{R}_p = R_p[\alpha]\) where \(\alpha\) is an eigenvalue of Frobenius. Fix a weight \(k \ge 2\). If \(R^{\mathrm{ord}}_p\) denotes the ordinary deformation ring, then there is a corresponding quotient \(\widetilde{R}^{\mathrm{ord}}_p\) of \(\widetilde{R}_p\) which records ordinary deformations together with the action of Frobenius on the unramified quotient. The \(\overline{\mathbf{Q}}_p\)-points of these local deformation rings are the same (since \(k \ge 2\)). The usual Taylor-Wiles-Kisin method produces an isomorphism of the type \(\widetilde{R}[1/p] = \widetilde{\mathbf{T}}[1/p]\), where \(\widetilde{R}\) is the global deformation ring which takes into account the extra data at \(p\). This isomorphism holds for all weights \(k \ge 2\), which is enough to get an isomorphism on the corresponding ordinary families.
If \(\rho(\mathrm{Frob}_p)\) has distinct eigenvalues, one may now deduce Buzzard-Taylor (by the same argument as BT). If \(\rho(\mathrm{Frob}_p)\) is scalar, then one has to make a slight adjustment. To see what to do, note that if \(f(\tau)\) is the desired weight one form, then the old space \(f(\tau), f(p \tau)\) can no longer be diagonalized with respect to \(U_p\). Instead, it should give rise to a surjective map \(\psi: \widetilde{\mathbf{T}} \rightarrow \mathcal{O}[\epsilon]/\epsilon^2\) such that the image of \(\mathbf{T}\) is \(\mathcal{O}\). Conversely, if there is such a map \(\psi\), this produces the ordinary old forms necessary to recover \(f\) (both forms have the same Hecke eigenvalues away from \(p\), so one can determine the \(q\)-expansions). From the modularity result, it suffices to construct such a map on the Galois side. Yet this exists precisely because \(\rho\) comes with two distinct unramified quotients.
DG and I used these these flavours of deformations rings for a somewhat different purpose (although we required more precise integral information concerning \(\widetilde{R}^{\mathrm{ord}}_p\) coming out of a very nice paper of Snowden). On the other hand, as far as the argument above goes, it was apparently known to RLT many years ago (as I learnt by chatting with TG whilst drinking an $8 can of Boddingtons in Toronto).
Suppose one assumes instead that \(\overline{\rho}\) is reducible. Recall that one has maps \(R^{\mathrm{ps}} \rightarrow R\) and \(R^{\mathrm{ps}} \rightarrow \mathbf{T}\) for a suitable pseudo-deformation ring \(R^{\mathrm{ps}}\). In higher weights, Skinner-Wiles essentially prove that \(R[1/p]^{\mathrm{red}} = \mathbf{T}[1/p]\), which should be sufficient to construct the required overconvergent forms \(f_{\alpha}\) and \(f_{\beta}\) in weight one. While chatting with PA over espresso today, it also seems reasonable that (using appropriate framings, as above) one may generalize this to the case where \(\overline{\rho}\) is no longer \(p\)-distinguished, as long as the characteristic zero eigenvalues \(\alpha\) and \(\beta\) are distinct. The problem, however, with the \(\alpha = \beta\) case is that one needs to promote a non-reduced quotient \(R \rightarrow \mathcal{O}[\epsilon]/\epsilon^2\) to a map from \(\mathbf{T}\), and the methods of Skinner-Wiles in the reducible case only give information about the reduced quotients of \(R\). Is there any way around this?
This seems (pretty close) to the only remaining obstruction for a complete solution to the weight one odd case of Fontaine-Mazur.
