Let $G$ be the absolute Galois group of a finite extension of $\mathbb{Q}$ or $\mathbb{Q}_p$. Let $k$ be a perfect field of characteristic $l>0$ (possibly $l=p$).

If we have a homomorphism $G\rightarrow GL_n(k)$ for some $n>0$, can we lift it to a homomorphism $G\rightarrow GL_n(\mathbb{W}(k))$? What if we require the lift to be continuous (with respect to the cofinite group topology and $p$-adic topology)? If the answer is no, are there any non-liftable representations of "geometric nature" (i.e. something of this sort)?


Let's consider the case when $k$ is a finite field and $G=\text{G}_{\mathbb{Q}}$. When $n=1$, you can take the Teichmuller lift, so assume that $n>1$. The first geometric lifting theorems were proved by Ramakrishna for odd residual representations to $\text{GL}_2$ which are irreducible and unramified at finitely many primes.

The state of the art on Geometric lifting theorems when the residual representation is irreducible is due to Fakhruddin, Khare and Patrikis "Relative deformation theory and lifting irreducible Galois representations" https://arxiv.org/abs/1904.02374, this introduces a very new technique (informally dubbed "the doubling method") in the deformation theory as a result of which many assumptions on prior lifting theorems can be removed.

No, it is not true that you can lift any residual Galois representation to a geometric one. If n=2 and your residual Galois representation is even (ie the determinant character evaluated on complex conjugation is trivial) then one would expect it to come from an algebraic Mass form, in particular the lift would be expected to have finite image. Calegari has an argument to show that indeed this heuristic is true when l is sufficiently large.

For more general algebraic groups there is an analogue of the odd-ness condition, this requires the group to contain -1 in its Weyl group, this is more or less the same condition as the existence of a split Cartan involution (this is where complex conjugation should map to under the residual representation). There are no such odd residual representations to $\text{GL}_n$ for $n>2$. The odd-ness condition is necessary, therefore geometric lifting theorems do not apply to $\text{GL}_n$ for $n>2$.

For residually reducible representations $G\rightarrow GL_2(k)$ which are indecomposable and odd, it was shown by Hamblen and Ramakrishna (under suitable conditions) that it does lift to a geometric Galois representation. It follows from the work of Skinner and Wiles "Residually Reducible Galois Representations and Modular forms" that such representations arise from cuspidal Hecke eigenforms. This result was generalized by myself to the representations $G\rightarrow \text{GSp}_{2n}$ to obtain geometric lifts, these are not known to be automorphic as the only higher dimensional generalization of the results of Skinner and Wiles which is due to Thorne is for residual representations for Galois representations over CM number fields.

In principal there is no reason to expect that residual representations which are simply a sum of characters should not lift to characteristic zero representations coming from Hecke eigencuspforms, examples of such can be found in the paper of Skinner and Wiles after a certain totally real base change (you'll have to dig through the details to see this).

Another issue with these lifting theorems is that normally we work with representations which are crystalline at p, the results of Kisin has been vastly generalized to as to make this possible in considerable generality (and it would be of interest to generalize this to more general p-adic Hodge theoretic conditions for larger groups).

Moving on to finding continuous lifts which are not necessarily geometric, if you allow infinite ramification of your lift, it is possible to lift irreducible Galois representations $G\rightarrow\text{GL}_n(k)$ to continuous Galois representations $G\rightarrow GL_n(W(k))$. See Ramakrishna's paper "Infinitely ramified Galois representations" and subsequent papers. The key technique which is adapted here is due to Khare, Larsen and Ramakrishna. The set of primes of ramification for the lift will have density zero. Also, theorem 1.5 "Algebraic Monodromy groups of G-valued l-adic representations of $G_{\mathbb{Q}}$" in a recent paper of Shiang Tang could be of some interest.


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