I was wrong!
I have said on a number of occasions that my favourite family of groups for the inverse Galois problem is \(G=\mathrm{SL}_2(\mathbf{F}_p)\). This is still true, and the reason is still the same; the fact that \(G\) has no non-trivial non-central involutions means that any such extension must be totally real up to twist, and thus it cannot come (for example) as the mod-\(p\) reduction of some strongly compatible system (assuming \(p > 5\)). For a similar reason, the group \(\mathrm{PSL}_2(\mathbf{F}_p)\) should also present difficulties under the additional assumption that the field is totally real, and I used this to argue that a totally real extension with Galois group \(\mathrm{PSL}_2(\mathbf{F}_{23})\) would be more interesting to me than any other Galois closure of a degree \(24\) field. It was then a delight to receive an email from Eray Karabiyik (a recent student of Zywina from Cornell, soon to be a postdoc at HIMIS, Shenzhen) giving me an explicit degree \(24\) polynomial with all roots real whose splitting field had Galois group \(\mathrm{PSL}_2(\mathbf{F}_{23})\)!
Here is an introduction to his construction. The goal is to construct (more or less) a representation
\[\rho: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(\mathbf{F}_{23})\]
which is even, has image containing \(\mathrm{SL}_2(\mathbf{F}_{23})\), and whose determinant lands in the squares \((\mathbf{F}_{23}^{\times})^2\). Its projectivization then has image exactly \(\mathrm{PSL}_2(\mathbf{F}_{23})\). As mentioned above, this should not come from a regular rank \(2\) (with coefficients) motive. But there is no such parity obstruction to its appearing as a constituent of the mod-\(23\) reduction of a rank \(4\) motive! So, for example, one could hope to find an abelian surface \(A\) with
\[A[23]^{\mathrm{ss}} \simeq \rho \oplus \rho^{\vee}(1)\]
where \(\rho\) is even and \((1)\) denotes the mod-\(23\) cyclotomic twist. Suppose for now that \(A\) and a principal polarization are defined over \(\mathbf{Q}\).
One way to find \(A\) so that \(A[p]\) breaks up is for its geometric endomorphism ring to be the ring of integers \(\mathcal{O}_K\) of a real quadratic field \(K\). If this action is defined over \(\mathbf{Q}\), however, then one just gets (once again) rank \(2\) compatible systems which are odd. Suppose instead that the action of \(K\) is only defined over a quadratic field \(E\). Now, for a prime \(p\) that splits in \(K\), \(A[p]\) decomposes over \(E\) into two factors which are interchanged by the action of \(\mathrm{Gal}(E/\mathbf{Q})\). On the other hand, at an inert prime, one gets a two-dimensional representation of \(G_E\) over \(\mathbf{F}_{p^2}\), with the non-trivial coset of \(G_E\) acting semilinearly through the field automorphism, and generically with large image.
But if \(p\) is ramified in \(K\), with \((p)=\pi^2\), then the subspace \(A[\pi]\) does give a representation \(\rho:G_{\mathbf{Q}}\rightarrow\mathrm{GL}_2(\mathbf{F}_p)\). Writing \(\omega\) for the mod-\(p\) cyclotomic character and \(\chi\) for the quadratic character of \(E\), one obtains
\[0\longrightarrow\rho\longrightarrow A[p]\longrightarrow\rho\otimes\chi\longrightarrow0,\qquad \det\rho=\omega\chi.\]
Thus \(\rho\) is even iff \(E\) is imaginary. The condition that the determinant be a square is precisely \(\chi=\omega^{(p-1)/2}\) with \(p=23\). So one is indeed in with a chance assuming:
- \(K\) is a real quadratic field with discriminant divisible by \(23\),
- \(E=\mathbf{Q}(\sqrt{-23})\).
In this case \(\det\rho=\omega^{12}\) and \(\rho\otimes\chi\simeq\rho^{\vee}(1)\), as desired. Of course, large image still has to be checked.
To find surfaces \(A\) with endomorphisms by \(K\), it helps to have access to the corresponding Hilbert modular surface. One can also insert \(E\) into the story by suitable twisting. This is analogous to how one can look for \(\mathbf{Q}\)-curves defined over a quadratic field \(E\), admitting an isogeny of degree \(q\) to their conjugate, by twisting \(X_0(q)\) by the Atkin–Lehner involution and the character of \(E\). The quotient \(X_0^+(q)\) forgets which conjugate one started with.
Here Eray takes \(K=\mathbf{Q}(\sqrt{23})\), for which Elkies and Kumar give an explicit model of \(Y_-(92)\). There are two relevant involutions: one exchanges the two RM embeddings, while the other switches the two principal polarization classes on the generic unpolarized RM surface. The latter correspond to totally positive units modulo squares (the fundamental unit \(\epsilon\) is totally positive): starting with a principal polarization \(\lambda\) and RM embedding \(\iota\), the other class is represented by \(\lambda\circ\iota(\epsilon)\). Both leave the unpolarized surface unchanged, corresponding to “\(q=1\)”. Eray searches on the appropriate non-K3 involution quotient, twisted by \(E\) using the remaining involution. If I understood him, this turns out to be (birational to) a genus two fibration over \(\mathbf{P}^1\). He then finds a point which gives rise to such an \(A\).
Now I have simplified Eray’s example, because his \(A\) is defined over an auxiliary quadratic field \(H=\mathbf{Q}(\sqrt{18193})\), with the RM defined over \(EH\). This is presumably either related to field of moduli versus field of definition issues at low level, or (somewhat relatedly) lifting rational points from the quotient by the involution back to the cover. It affects some of the discussion of Galois representations above, but (in favorable situations) is no longer visible when one considers projective representations.
As \(p\) gets larger, one will either have to make \(K\) larger (so that \(p\) is ramified) or take a non-maximal order like \(\mathbf{Z} + p \mathcal{O}_K\). In either case, the corresponding Hilbert modular surfaces (and all their quotients by involutions) will have general type for large enough \(p\), and my guess is that rational points away from the special loci on the relevant Hilbert modular surfaces and their twisted quotients will become increasingly scarce and most likely eventually empty. So I suspect this is unlikely to produce totally real \(\mathrm{PGL}_2(\mathbf{F}_p)\) or \(\mathrm{SL}_2(\mathbf{F}_p)\) extensions for all (or even infinitely many) \(p\), but it’s a very nice example nonetheless.