## I don’t know how to prove Serre’s conjecture.

I find it slightly annoying that I don’t know how to prove Serre’s conjecture for imaginary quadratic fields. In particular, I don’t even see any particularly good strategy for showing that a surjective Galois representation — say finite flat with cyclotomic determinant for $v|p$

$\overline{\rho}: G_{F} \rightarrow \mathrm{GL}_2(\mathbf{F}_3)$

is modular of the right level. The first problem is that the strategy used by Wiles does not work. The results of Langlands-Tunnell imply the existence of an automorphic form $\pi$ for $\mathrm{GL}(2)/F$ which has an associated finite image Galois representation into $\mathrm{GL}_2(\mathbf{Z}[\sqrt{-2}])$ with projective image $A_4$ that is “congruent” to $\overline{\rho}$ modulo a prime above $3$, but there is no way to realize this congruence in cohomology. An analogous example over $\mathbf{Q}$ would be that the (known) modularity of a surjective even Galois representation:

$\overline{\rho}: G_{\mathbf{Q}} \rightarrow \mathrm{SL}_2(\mathbf{F}_4) = A_5$

has no implications for the modularity of the corresponding even complex representation with projective image $A_5$ (which is “congruent” modulo $2$), because there is no way to relate them via Betti or coherent cohomology.

One context in which we have a fairly satisfactory answer to Serre’s conjecture over imaginary quadratic fields is for representations $\overline{\rho}$ which are the restriction of an odd representation of $G_{\mathbf{Q}}$. (I guess one also has modularity in some CM cases, that is, inductions from CM extensions $H/F$.) So, if we give ourselves modularity lifting results (surely a requirement to get anywhere), one could imagine trying to play some sort of game using the $3$$5$ switch to construct a chain between a representation which comes from $\mathbf{Q}$ and the target representation. Or, perhaps, one can play the $3$$3$ game using abelian surfaces with real multiplication by $\mathbf{Z}[\sqrt{7}]$. However, there’s a big hole in this strategy: the $3$$5$ game presupposes that once you know that $\overline{\rho}$ is modular of some level, you know it at minimal level. So now one runs into the problem of level lowering. Alternatively, if you want to play the $3$$5$ game Khare–Wintenberger style, you really have to construct minimal lifts. But such lifts will not (in general) exist over imaginary quadratic fields.

This seems to be a serious problem. The only general strategies I can imagine involve being able to push the torsion classes around to different groups using some (as yet unknown) functoriality for torsion classes. (For example, find minimal lifts over some large CM extension $F'/F$, prove modularity over $F'$, and then invoke non-abelian base change for torsion classes to recover modularity of the original representation.) The other argument would be to examine the corresponding Eisenstein classes for $U(2,2)/F$. This seems a little fishy, however; one would really want to see these representations inside (etale) cohomology in order to invoke some kind of Mazur principle, but as we have noted previously, the Galois representations of interest don’t actually live inside the etale cohomology groups that one might want them to. Ultimately, the basic problem is that the classical (Mazur-Ribet) style arguments make strong use the geometry of modular curves (which is certainly missing here) and the more modern approaches (starting with Skinner–Wiles) rely on base change.

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