# Lifting SL2(k) to a subgroup of Witt vectors

$$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\W{W}$$Let $$k$$ be a finite field, and let $$\W_n(k)$$ be the degree $$n$$ Witt vectors over $$k$$ (so $$\W_1(k) = k$$).

Does there exist a subgroup $$H \le \SL_2(\W_2(k))$$ that maps isomorphically onto $$\SL_2(k)$$?

If $$k$$ has characteristic $$p \ge 5$$, no such subgroup exists. This follows from a well-known argument of Serre (in his "Abelian $$\ell$$-adic representations" book) which proves slightly more – any subgroup of $$\SL_2(\W_2(k))$$ which surjects onto $$\SL_2(k)$$ must be the whole group. It's also easy to check the claim by hand for $$k = \mathbf{F}_2$$, although Serre's stronger assertion does not work for $$\mathbf{F}_2$$.

On the other hand, for $$k = \mathbf{F}_3$$ such a pathological subgroup really does exist (coming from the 2-dimensional characteristic 0 representation of $$\GL_2(\mathbf{F}_3)$$ that is a vital input in the proof of Fermat's Last Theorem).

What happens for larger fields of characteristic 2 or 3?

I found the answer myself with the help of a very useful hint from user "nobody" in the comments, so I'm going to post a community-wiki answer in case anyone else finds it useful.

My question is answered by Manoharmayum in this paper:

Manoharmayum, Jayanta, A structure theorem for subgroups of (GL_n) over complete local Noetherian rings with large residual image., Proc. Am. Math. Soc. 143, No. 7, 2743-2758 (2015). ZBL1338.20047.

Lemma 3.7 of the paper confirms that $$SL_2(k)$$ lifts to a subgroup of $$SL_2(W_2(k))$$ if and only if $$k = \mathbf{F}_3$$.