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Q: What is the "simplest" finite group $G$ for which we don't know how to realise it as a Galois group over $\mathbf{Q}$ ?

So here the word simplest might be interpreted in a broad sense. If you want something precise you may take the group of smallest order but I prefer to leave the question as it is. Also since naturally one classifies finite groups into families one may also ask the following

Q: What is the "simplest" example of a family of finite groups for which the inverse Galois problem is unknown?

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Among the sporadic family, I think there is only one group for which the inverse problem is unknown-- $M_{23}$. See… – Pace Nielsen Jul 12 '11 at 17:46
An almost duplicate of… ? – Mark Sapir Jul 12 '11 at 19:03
Thanks Mark for the link – Hugo Chapdelaine Jul 12 '11 at 19:37

I am not an expert and I might be misrembering some talks I've attended. Anyway, $SL_2(\mathbb{F}_q)$ can be done for prime $q$ by using torsion on non-CM elliptic curves. But I don't think it's been done for general prime powers $q$. Also, what about $SL_3$?

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The $p$-torsion on a generic elliptic curve gives not ${\rm SL}_2({\bf F}_p)$ but ${\rm GL}_2({\bf F}_p)$ with cyclotomic character. In many cases this can be twisted to get ${\rm SL}_2({\bf F}_p)$, but it's not immediate. Serre discusses this in some detail in his Topics in Galois Theory (1992). – Noam D. Elkies Jul 13 '11 at 6:13
For general powers of q, there are some results (which do not cover all cases). In and in they prove that $PSL_2(\mathbb{F}_q)$ and $PGL_2(\mathhbb{F}_q)$ is the Galois group of an extension for almost al $q$ (in the second work they prove that fixed n, in a density one set of primes $\ell$, $PSL_2(\mathbb{F}_{\ell^n})$ is obtained). Then you can just lift the representation to get a rep. of SL_2 if you want. – A. Pacetti Jul 13 '11 at 7:59
Yes, it's ${\rm PSL}_2({\bf F}_p)$, sorry. Chapter 5 of Serre starts with the construction of a regular extension of ${\bf Q}(T)$ with Galois group ${\rm PSL}_2({\bf F}_p)$ if 2, 3, or 7 is a quadratic nonresidue of $p$ [K-y. Shih, Math.Ann. 207 (1974) and Comp.Math. 36 (1978)]. At the end of Chapter 5 he says regular extensions of ${\bf Q}(T)$ with Galois group ${\rm SL}_2({\bf F}_q)$ had been constructed for $q\leq 9$, but at the time he didn't know of one with $q=11$ or higher, and that Mestre found ${\rm SL}_2({\bf F}_{2^m})$ over ${\bf Q}$ for $m\leq 16$ with modular forms. – Noam D. Elkies Jul 14 '11 at 0:17
I have another issue. At no point is there anything about "almost all primes $\ell$" or even a "density one set of primes" in the paper of Dieulefait and Wiese. Rather, fixing either a prime $\ell$ and varying $n$ or fixing $n$ and varying $\ell$ there is a positive density of $n$ or $\ell$ where $\mathrm{PSL}_2(\mathbf{F}_\ell)$ is a Galois group. As it stands, what we know is that Shih's Theorem gives that for $(7/8)$ of the primes, $\mathrm{PSL}_2(\mathbf{F}_p)$ is a Galois group and considerably more by work of Clark if we believe BSD. – stankewicz Jul 14 '11 at 15:40
@Amy, you are right that in general you have a condition to lift a representation (an element in some H^2), but in this case, the representations they construct are modular, so in particular do lift. And it is true that they do not prove a density 1 result, but a positive density result... – A. Pacetti Jul 18 '11 at 10:45

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