Inverse Galois Problems II

David Zywina was in town today to talk about a follow up to his previous results mentioned previously on this blog. This time, he talked about his construction of Galois groups which were simple of orthogonal type, in particular, the simple groups

\(\Omega(V) \subset \mathrm{SO}(V) \subset \mathrm{O}(V)\)

where \(V\) is a vector space \(V\) over \(\mathbf{F}_l\) of odd dimension at least five. The group \(\Omega(V)\) here is a simple group of index two inside \(\mathrm{SO}(V).\) In the special case when \(n = 5,\) there is an exceptional isomorphism

\(\Omega(V) \simeq \mathrm{PSp}_4(\mathbf{F}_l).\)

In constrast to his constructions of number fields with Galois group \(\mathrm{PSp}_2(\mathbf{F}_l),\) Zywina actually constructs a family of compatible families whose residual image is generically \(\Omega(V).\) When David told me about this construction (scribbled on a piece of paper) in Frankfurt airport on the way back from Oberwolfach, I was troubled by something which I shall now explain. Without saying so much about the construction (you can read about it here), the compatible families of Galois representations of interest occur inside \(H^2(X_T,\mathbf{Q}(1))\) for a carefully chosen family of non-isotrivial elliptic sufaces \(X.\) As David explained in his talk today, the zero section and the fibres of bad reduction contribute a large Galois trivial summand to \(H^{1,1},\) and the remaining piece is five dimensional. What disturbed me at the time was that this construction was surely liftable to a compatible family of four dimensional representations with generalized symplectic image. After all, Tate’s result on \(H^2(G_{\mathbf{Q}},\mathbf{C}^{\times})\) guarantees that one can lift any projective representation with image in \(\mathrm{PSp}_4(\mathbf{F}_l)\) to a genuine generalized symplectic representation. This representation should then come from a Siegel modular form, since all oddness conditions should be automatic. On the other hand, if you want a family of Galois representations giving rise to a family of Siegel modular forms, especially one for which the maximal difference between any two Hodge-Tate weights in \(\wedge^2 W\) is two, then you expect that they have to come from a family of abelian surfaces, or at least abelian varieties \(A\) of dimension \(2n\) with endomorphisms by the ring of integers in a totally real field of degree \(n.\) However, there is an obstruction to making this work — the corresponding Galois representations will have Hodge-Tate weights \([0,0,1,1],\) and they will have similitude character that is an even finite order character times the cyclotomic character. It’s easy to see that for such a family, the residual representations will (at least half the time) land in \(\mathrm{PGSp}_4(\mathbf{F}_{l})\) and not in the simple index two subgroup, similar to what happens for modular forms of weight two. I thought at the time that I must have been making some group theory error, so after today’s talk we sorted out the details:

In the process of this computation, however, I realized what my error actually was. I was imagining that the original compatible family of Galois representations in \(H^2\) had Hodge-Tate weights \([0,1,1,1,2],\) but they could equally have had Hodge-Tate weights \([0,0,1,2,2].\) And in this latter case, the Galois representation (up to twist) of the corresponding Siegel modular form in \(\mathrm{GSp}_4\) will have Hodge-Tate weights \([-1,0,0,1].\) In particular, we are not looking for classical Siegel modular forms of low weight, but the nasty Siegel modular forms which do not contribute to holomorphic limits of discrete series and only occur in coherent cohomology via \(H^1\) or \(H^2.\) (A reference for this fact is George Boxer’s talk in Barbados.) And now everything makes sense! That is, if you have a family of Galois representations with Hodge Tate weights \([-1,0,0,1]\) and quadratic similitude character, then (with some good luck) you can really have projective representations which land in the right simple group for all but finitely many \(l.\)

A related point: when lifting projective representations using Tate’s theorem, one may have to increase the size of the residue field. In fact, when \(\ell \equiv 3 \mod 4,\) it will not be possible to lift an odd \(\mathrm{PSp}_4(\mathbf{F}_l)\) representation to one in \(\mathrm{GSp}_4(\mathbf{F}_l)\) (there is an obstruction at infinity). Indeed, the natural lift is the group \(\mathrm{Sp}_4(\mathbf{F}_l)\) together with a scalar matrix \(I\) with \(I^2 = -1.\) This suggests what the picture should be motivically: there should be an eight dimensional piece of \(H^2(Y)\) (for some \(Y)\) which admits an involution breaking the representation up into two four dimensional pieces, and these pieces will have coefficients in \(\mathbf{Q}(\sqrt{-1}).\) Can one find such a \(Y\) explicitly? This does remind me of the motivic lifting problems that Stefan Patrikis knows about.

From this analysis, it also becomes clearer why Zywina could find a family of compatible families with residual image \(\Omega(V)\) when \(\mathrm{dim}(V)\) is odd and at least five, but only isolated examples of compatible families with residual projective image \(\mathrm{PSL}_2(\mathbf{F}_l) \simeq \Omega(V)\) with \(\mathrm{dim}(V) = 3.\) In the latter case, the corresponding modular forms will be forced to have odd weight \(k > 1,\) and so the Hodge-Tate weights will differ by at least two, and so Griffiths’ theorem implies that they should not deform in a family. On the other hand, if you want to look for Siegel modular forms which could possibly correspond to geometric families, and you want the similitude character to be an even power of the cyclotomic character times a finite character, then it is possible to escape the specter of Griffiths on your shoulder, but only barely — you will be pretty much forced to work with forms whose HT weights are \([-1,0,0,1].\) Of course, I’m not sure I can prove that any Siegel modular forms of this kind actually exist! (insisting the Mumford-Tate group is big, naturally). My proposal in the previous post to look for these representations using Siegel modular forms would also have only found sporadic compatible families, because to ensure computability and the determinant condition I suggested looking in weights where the Galois representation was regular and had HT weights something like \([0,1,3,4],\) — the gap being necessary to make the multiplier character a square of a Hodge-Tate character.

There is one check left on these musings (though I’m sure it must be correct), namely, that for the surface \(X\) in 1.4 of Zywina’s paper, one should have

\(h^{2,0}(X) = 2.\)

Update: Here’s a proof of this statement:

Proof: The Hodge diamond of a minimal elliptic surface \(\pi: X \rightarrow C\) was computed by Miranda, see IV.1.1 here; I’ll try to give a self contained argument. Let \(\omega_E\) be the Euler characteristic of \(\mathcal{O}_X.\) Let \(L^{-1}\) be the bundle \(L^{-1} = R^1 \pi_* \mathcal{O}_X\) on \(X;\) it is a line bundle because the fibres are elliptic curves, so it makes sense to talk about \(L.\) The bundle \(L\) has positive degree if and only if the fibration is not isotrivial (this is not so hard, but let me give the proof of III.1.6 of Miranda as a reference); let us assume this is the case. From the Leray spectral sequence, there is an exact sequence

\(0 \rightarrow H^1(C,\pi_* \mathcal{O}_X) \rightarrow H^1(X,\mathcal{O}_X) \rightarrow H^0(C,L^{-1}) \rightarrow H^2(C,\pi_* \mathcal{O}_X)\)

Since \(\pi_* \mathcal{O}_X = \mathcal{O}_C,\) the first term has dimension \(g,\) the genus of \(C.\) Since we are assuming that \(L\) has positive degree, the third term is also zero, and hence the irregularity of a non-isotrivial elliptic surface is

\(H^1(X,\mathcal{O}_X) = g.\)

It follows that

\(\chi(\mathcal{O}_X) = h^{0,0} – h^{1,0} + h^{2,0} = 1 – g + h^{2,0}.\)

In our particular case, the genus of \(C\) is zero. On the other hand, as noted in 2.4 of Zywina’s paper, the degree of the minimal discriminant is \(12 \cdot \chi_E = 12 \cdot \chi(\mathcal{O}_X).\) In the example at hand, Zywina computes (see section 8) that \(\chi_E = 3,\) and so

\(h^{2,0} = 3 – (1-g) = 3 – 1 = 2.\)