Today’s post is about work of my student Andreea Iorga!
A theorem of Ozaki from 2011, perhaps not as widely known as expected, says the following:
Theorem: Let \(p\) be prime, and let \(G\) be a finite \(p\)-group. Then there exists a number field \(F\) and an extension \(H/F\) such that:
- \(H/F\) is the maximal pro-\(p\) extension of \(F\) which is everywhere unramified.
- \(\mathrm{Gal}(H/F) = G\).
Since any non-trivial \(p\)-group \(G\) has a non-trivial center, it can be written as a central extension of a smaller \(p\)-group \(G’\) by \(\mathbf{Z}/p \mathbf{Z}\), and thus the proof is (as one might imagine) by induction. The structure of the argument is quite tricky, and it’s a little hard to absorb all the ideas at once. There is a new preprint by Hajir, Maire, and Ramakrishna which gives both a simplification and also an extension of Ozaki’s result (the extension being that one has more explicit control over the degree of \(F\)).
But this post will actually be about a somewhat different generalization due to my student Andreea Iorga (details currently being written up!). Let me give her result now:
Theorem [Iorga] Let \(\Phi\) be a finite group of order prime to \(p\), and let \(G\) be a finite \(p\)-group with an action of \(\Phi\). Assume there exists an extension \(L/K\) such that:
- \(L/K\) is Galois with Galois group \(\Phi\),
- \(K\) contains \(\zeta_p\),
Then there exists number fields \(H/F/E\) such that:
- \(H/F\) is the maximal pro-\(p\) extension of \(F\) which is everywhere unramified.
- \(\mathrm{Gal}(H/F) = G\).
- \(\mathrm{Gal}(H/E) = \Gamma\), where \(\Gamma\) is the semi-direct product of \(\Phi\) by \(G\) corresponding to the given action of \(\Phi\) on \(G\).
When \(\Phi\) is trivial, one recovers Ozaki’s theorem in the case when \(p\) is a regular prime. In fact, Ozaki’s first proof also has a similar hypothesis. Most likely Iorga’s argument extends to the more general case where one does not need to assume that \(\zeta_p \in E\). (Of course, in order not to accidentally solve the inverse Galois problem, the other two conditions on \(L\) and \(K\) will be necessary!)
One nice consequence (and a motivating example) of Iorga’s theorem is as follows. Consider absolutely irreducible residual representations:
\(\overline{\rho}: G_{K} \rightarrow \mathrm{GL}_2(\mathbf{F}_p)\)
to a finite field. What possible rings \(R\) can occur as deformation rings of all such \(\overline{\rho}\)? In this setting, let \(R\) denote the deformation ring of everywhere unramified representations. Let’s also assume that the image (to be absolutely concrete) has order prime to \(p\), say projectively \(\Phi = A_4\) or \(S_4\). The Fontaine–Mazur conjecture predicts that the only \(\overline{\mathbf{Q}_p}\)-points will have finite image, and thus correspond to the natural lift (assuming the characteristic of \(k\) is \(p \ge 5\)). An argument with class groups then implies that one should expect \(R[1/p] = \mathbf{Q}_p\), or equivalently that \(R\) is a ring admitting a map
\(R \rightarrow \mathbf{Z}_p\)
with finite (as a set) kernel \(I\). A consequence of Iorga’s theorem is the following:
Theorem [Iorga] Let \(R\) be any local ring admitting a surjection to \(\mathbf{Z}_5\) with finite kernel. Then \(R\) is a universal everywhere unramified deformation ring.
This is true for more general regular primes \(p \ge 5\) under the further assumption on the existence of an \(A_4\)-extension of \(\mathbf{Q}(\zeta_p)\) with class number prime to \(p\), the point is that one can find such an extension explicitly for \(p=5\). The key idea to reduce this theorem to the previous one is a follows. Suppose that the image of \(\overline{\rho}\) is \(\widetilde{\Phi}\). Since this has order prime to \(p\), it lifts to a representation \(\widetilde{\Phi} \subset \mathrm{GL}_2(\mathbf{Z}_p)\). Then let \(\Gamma\) denote the inverse image of this group inside \(\mathrm{GL}_2(R)\), so it lives inside an exact (split) sequence:
\( 1 \rightarrow 1 + M_2(I) \rightarrow \Gamma \rightarrow \widetilde{\Phi} \rightarrow 1\)
The group \(\Gamma\) admits a natural residual representation via \(\overline{\rho}\), and clearly \(\Gamma\) admits a deformation to \(\mathrm{GL}_2(R)\) by construction. The point is that one can show that this \(R\) is the universal deformation ring, and hence providing one has extensions \(H/F/E\) with \(\mathrm{Gal}(H/E) = \Gamma\) and \(H/F\) the maximal everywhere unramified pro-\(p\) extension of \(F\) (using the previous theorem) one is in good shape. (There is a trick to reduce the problem in this case to \(\Phi\) in order to make the “base case” easier, since one has fields \(F/\mathbf{Q}\) and \(\widetilde{F}/\mathbf{Q}\) with \(\mathrm{Gal}(F/\mathbf{Q}) = \Phi\) and \(\mathrm{Gal}(\widetilde{F}/\mathbf{Q}) = \widetilde{\Phi}\) and if \(\Phi = A_4\) and \(p = 5\) then proving that \(F(\zeta_5)\) of degree \(48\) has class number prime to \(5\) is easier than the same claim for \(\widetilde{F}(\zeta_5)\) of degree some multiple of \(96\)).
One way to view this result is as an example of “Murphy’s Law” for moduli spaces. This is the idea explained by Ravi Vakil that all possible singularities occur inside deformation spaces (of a more geometric kind rather than Galois deformation rings). The analogue in the setting of Galois deformation rings is to say that all possible local rings (subject to some obvious constraints) occur as Galois deformation rings. Still considering the case of everywhere unramified deformation rings, another natural class of rings one might expect to arise in this way is the set of all finite artinian local rings. Of course for such rings one would have to consider residual representations whose images have order divisible by \(p\), requiring a further modification of the theorems of Ozaki and Iorga. In a different direction, one can ask what happens for deformation conditions with other local conditions at \(p\). Here are two natural such questions:
Problem Let \((R,\mathfrak{m})\) be any complete local Noetherian ring with finite residue field which is finite over \(W(k)\). Then does \(R\) occur as the finite flat deformation ring of some absolutely irreducible residual representation?
Problem Let \((R,\mathfrak{m})\) be any complete local Noetherian ring with finite residue field over \(W(k)\), and assume that:
- \(R\) is a complete intersection, namely that there is a presentation:
\(R \simeq W(k)[[x_1,\ldots,x_d]]/(f_1,\ldots,f_r)\)
where \(d \ge r\).
- \(p \in R\) is a regular element.
Then does \(R\) occur as the universal deformation ring (with fixed determinant) of some
absolutely irreducible residual representation? Note that the conditions given are both conjectured (but unknown in general) to be necessary conditions in this case.
I would guess the first problem has a positive answer but I’m honestly not even sure about the second one! This is already very interesting in the case (say) of a totally even representation with the addtional requirement that \(r = d\).
Update: A friend of the blog points out that the second problem most likely falls prey to countability issues when \(R\) is not finite over \(W(k)\) and indeed that seems to be an issue. I’m not quite sure what the optimal modified version should be; perhaps one could ask that for any \(k\) there are a deformation rings \((S,\mathfrak{m}_S)\) such that \(R/\mathfrak{m}^k = S/\mathfrak{m}^k_S\), perhaps even insisting that \(S\) is a complete intersection of the same dimension as \(R\) as well. The case when \(r=d\) still might be OK
