# Do local Galois representations always lift?

Suppose $G:=G_F$ is the absolute Galois group of a local (residue char. $\ell$) or global field $F$, and $\bar{\rho}$ a (linear) representation of $G$ on the $\mathbb{F}_q$-module $\mathbb{F}^d_q$, a 'residual Galois representation' in standard parlance. Suppose $\mathcal{O}$ is a characteristic (0,p) DVR with residue field $\mathbb{F}_q$. A lift of $\bar{\rho}$ is a representation $\rho$ on $\mathcal{O}^d$ reducing to $\bar{\rho}$ modulo the maximal ideal.

Very general question: When does a lift exist?

Some known cases I found: There is a lift when $F$ is global and

• d = 1 (Teichmüller) or
• $F$ is a function field (Böckle-Khare) or
• d=2, $\rho$ reducible (Khare) or
• $F=\mathbb{Q}$ (Ramakrishna, d=2) + (Hamblen, d>2)
• the obstruction to lifts is $Z^2(G,ad \bar{\rho})$, so if we assume things like $F\neq F(\mu_p)$ or $\bar{\rho}$ 'geometric' we can get R=T-style machinery running

If $F$ is local, then the structure of the universal lifting rings is sometimes well understood away from $p$: the generic fibers (if non-empty) are generically smooth and >0 dimensional if

• $\ell \neq p$ (Gee-BLGGT?)
• lots of subclasses of deRham (Kisin, among others)

Stupid but concrete question: Do these results imply, that there are nontrivial points in the generic fibers?

Somehow, I wasn't able to find the conditional 'if non-empty' in these papers. So, I'm wondering, if the existence of a lift in the local case is always "for free".

• You should ask this on overflow. – Alex Youcis Apr 29 '16 at 20:13

## 1 Answer

The existence of local lifts is open in general, even for $F = \mathbf{Q}_p$ and general $n$. One expects the stronger result that there exist de Rham lifts.

There is recent work of Gee-Hezig-Liu-Savitt on this problem (https://cms.math.ca/Events/winter15/abs/pdf/ant-fh.pdf). However, I do know some people who have an idea to answer the general case. I can get my people to talk to your people if you want.