## Local crystalline deformation rings

I just returned from a very pleasant conference in Puerto Rico courtesy of the Simons Foundation (general advice: if you live in Chicago, always accept invitations to conferences in January). One thing I learnt from Toby Gee was the following nice observation. Suppose that

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

is a modular Galois representation, which for convenience we shall assume is unramified outside $p$. Consider deformations of this representation which are crystalline with fixed Hodge-Tate weights $[0,k-1]$ where $k$ is even. According to Kisin, the global minimal crystalline deformation ring contains a point on every component of the corresponding local crystalline deformation ring. (All discussions of components refer to the generic fibres.) One natural question is how many components the local deformation rings actually have (when the weight is very small, it’s usually the case that there is only one such component and it is smooth — this was crucial in the original Taylor-Wiles method before Kisin). For higher weight, one can distinguish between components which are “ordinary” and “not ordinary”, but it is not clear what else there is. (Indeed, Kisin seemed to think some years ago that this would be it, using the meta-argument that amongst any finite set one should be able to distinguish different points by some naturally available property.)

Now suppose we also now assume that $\overline{\rho}$ is locally reducible. According to Buzzard’s conjectures, all the slopes of the global crystalline lifts of $\overline{\rho}$ will be integral. Suppose one wants to prove this by local methods. Then one is ultimately led to conjecturing that each component of the local crystalline deformation ring has a fixed integral slope (recall we are in the locally reducible case, this is certainly false for locally irreducible representations in general). As a first consequence, one sees that in very high weights there will be many different components. Moreover, if one takes a different
global representation $\overline{\varrho}$ which is the same locally as $\overline{\rho}$, then the set of slopes arising from lifts of $\overline{\varrho}$ will be the same as for $\overline{\rho}$. These ideas do not quite give a complete conjectural explanation of why Buzzard’s slope conjectures are true, but it is a good start.

Something that is a little disturbing in this picture, however, is the case when $\overline{\rho}$ is reducible. It becomes clear that, in high weight, there will be many crystalline representations with reducible residual representations, but the set of components of local crystalline deformation space which have a global point will be a proper subset of the set of components (assuming that components can be distinguished by slope).
For example, all the slopes at level one when $p = 2$ are (besides the Eisenstein series) $\ge 3,$ but there certainly exist modular forms of higher tame level with the same local residual representation of slope one. So is there any way to predict when a reducible representation will have a global lift on any component of local deformation space?

In fact, the failure of lifts in the reducible case is an old problem. In the most naive sense, one can find reducible representations at levels where there are no cusp forms, but to play the game honestly we should also allow (globally) reducible lifts. Perhaps the first genuine example corresponds to extensions:

$1 \rightarrow \mathbf{Z}/p\mathbf{Z} \rightarrow V \rightarrow \mu_p \rightarrow 1$

where the extension is completely split at $p$ but ramified at an auxiliary prime $N$. These representations are locally split and so certainly admit local lifts (namely, $\mathbf{Z}_p \oplus \mathbf{Z}_p(1)$). If $p > 3,$ then such extensions exist whenever $N \equiv \pm 1 \mod p$, but (by Mazur) one knows that there exist weight two level $\Gamma_0(N)$ lifts only when $N \equiv +1 \mod p$ (in fact, one can prove the analogous claim that there only exist global crystalline lifts with the appropriate conductors under the same congruence condition). This is related to the general problem of understanding when certain reducible representations can be lifted to cusp forms, which seems to be a tricky problem (Ken Ribet’s student Hwajong Yoo has thought about this).

This also reminds me of a fact I learnt from Kevin Buzzard. Take the representation

$\rho: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(\mathbf{Q}_2)$

associated to the cusp form $\Delta$. Then there exist lattices for this representation such that the corresponding residual representation is any one of the four (three non-split) extensions of $\mathbf{Z}/2\mathbf{Z}$ by itself which are unramified outside $2$. (Question: does this immediately imply the same is true for all $2$-adic representations coming from level one modular forms?)