Polymath Proposal: 4-folds of Mumford’s type

Let \(A/K\) be an abelian variety of dimension \(g\) over a number field. If \(g \not\equiv 0 \bmod 4\) and \(\mathrm{End}(A/\mathbf{C}) = \mathbf{Z}\), then Serre proved that the Galois representations associated to \(A\) have open image in \(\mathrm{GSp}_{2g}(\mathbf{Z}_p)\). The result is not true, however, when \(g=4\), as first noted by Mumford (in this paper).

The goal of this polymath project is to find an “explicit” example of such a Mumford \(4\)-fold over \(\mathbf{Q}\). There are a number of things I have in mind for what “explicit” might mean (this is, after all, supposed to be a polymath project so I’m not supposed to know how to do everything). But here is one way: associated to \(A\) is a compatible family of Galois representations

\(\rho_p: G_{\mathbf{Q}} \rightarrow \mathrm{GSp}_8(\mathbf{Z}_p)\)

such that, for some integer \(N\), the Galois representations \(\rho_p\) are unramified outside \(Np\), and for all other primes \(q\) the characteristic polynomial of \(\rho_p(\mathrm{Frob}_q)\) is equal to

\(Q_q(T) \in \mathbf{Z}[T]\)

for some polynomial which does not depend on \(p\). Then for example one could hope to give a list of the polynomials \(Q_q(T)\) for a collection of primes \(q\).

Here is the strategy to find such Galois representations.

We start by choosing a totally real cubic field, which for reasons to possibly be explained later should perhaps be \(F = \mathbf{Q}(\zeta_7)^{+}\). (One reason: it is the Galois cubic field of smallest possible discriminant.)

Step I: Find a Hilbert modular form over \(F\) of weight \((1,1,2)\) with coefficients in \(F\).

The idea here will be to follow the strategy employed by Moy-Specter (following Schaeffer) to compute a Hilbert modular form of weight \((1,3)\) over the field \(\mathbf{Q}(\sqrt{5})\). Namely, Let \(W\) denote the space of Hilbert modular forms of weight \((2,2,3)\) of some fixed level. Now divide by some suitable Eisenstein series of weight \((1,1,1)\) to get a space \(V\) of meromorphic forms of weight \((1,1,2)\). This will contain the (possibly zero) space \(U\) of holomorphic forms of weight \((1,1,2)\). The holomorphic forms will be preserved under the action of Hecke operators whereas \(V\) in general will not be. Hence one can start computing the intersection of \(V\) with its Hecke translates, which will also contain \(U\). Either you eventually get zero, or you (most likely) end up with an eigenform which you can hope to prove is holomorphic by proving its square is holomorphic.

Some Issues: The way that Moy-Specter compute the (analogue) of \(W\) is to use Dembélé’s programs to compute the Hecke eigensystems of that weight, and then use the fact that \(q\)-expansions are determined by the Hecke eigenvalues for Hilbert modular forms (suitably interpreted, one has to compute spaces of old forms of lower level etc.). The same idea should certainly work, but note that we are working here in non-paritious weight (that is, not all weights are congruent modulo \(2\)). My memory is that the current programs on the contrary assume that the weight is paritious. This would have to be fixed! Perhaps this is an opportunity for someone to code up Dembélé’s algorithms in sage?

Step II: Suppose one finds such a form \(\pi\). Note that I am also insisting that the coefficient field be as small as possible, namely the field \(F\) itself. Even though \(\pi\) is of non-paritious weight, there are still associated Galois representations (Some relevant references are this paper of Patrikis and also this paper of Dembélé, Loeffler, and Pacetti). More precisely, there are nice projective Galois representations, and these lift to actual representations, but they will not be Hodge-Tate; rather, up to twist (making the determinant have finite order, for example), they will have Hodge–Tate weights \([0,0]\), \([0,0]\), and \([-1/2,1/2]\). But now consider the tensor induction (twisted by a half!) of this representation from \(G_F\) to \(G_{\mathbf{Q}}\), that is, for \(\sigma \in \mathrm{Gal}(F/\mathbf{Q})\), the representations

\(\varrho:=\rho(\pi) \otimes \rho^{\sigma}(\pi) \otimes \rho^{\sigma^2}(\pi)(1/2)\)

Now these representations will be crystalline with Hodge-Tate weights \([0,0,0,0,1,1,1,1]\). Moreover, they will be symplectic, have cyclotomic similitude character, and (this is where the assumption on the coefficients of \(\pi\) comes in) will also have Frobenius traces in \(\mathbf{Q}\). OK, I literally have not checked any of those statements at all, but it kind of feels like it has to be true so that’s what I’m going with. The point of insisting that the coefficients of \(\pi\) was just \(F\) is to make the coefficients of this new system in \(\mathbf{Q}\). But this means (at least conjecturally) that these Galois representations have to come exactly from an abelian variety of Mumford’s type, because the Galois representations tell you that the Mumford-Tate group has Lie algebra \((\mathfrak{sl}_2)^3\).

Step III: Find this family in a different way. One issue with the construction above is that the Galois representations are not obviously motivic (or even satisfy purity!), so they certainly don’t provably come from an abelian variety. But it might be easier to find the actual variety once one knows its exact level. I’m not quite sure what I mean by “find” here — it’s an open question as to whether these Mumford 4-folds are Jacobians so I’m not entirely sure what one should be looking for.

Step IV: Bonus: prove that these \(4\)-folds have \(L\)-functions with meromorphic continuations (at least for \(H^1\) but it’s worth checking the other degrees as well) using triple product \(L\)-functions.

Some Further Remarks: There are a number of relevant papers by Rutger Noot that one should be aware of (An particularly relevant example: this one). There are restrictions on the possible level structures that can arise for Hilbert modular forms of this weight (in particular, they can’t be Steinberg at some place), so make sure not to waste time computing at those levels. This is related to the fact that the corresponding Shimura variety is compact. The actual associated Shimura variety is isomorphic to \(\mathbf{P}^1\) over the complex numbers; there’s some discussion in section 5.4 of Elkies’ paper. These Shimura curves naturally have models over the reflex field, which is \(F\) in this case, but actually they can sometimes be defined over even smaller fields, such as \(\mathbf{Q}\). Now I confess I am confused by a number of points, in increasing order:

1. What is the exact relationship between the model of this Shimura curve over \(\mathbf{Q}\) and the moduli problem? This is an issue both with understanding the moduli problem but also (because of the stackiness issues) differences between fields of moduli and fields of definition.
2. Does this Shimura curve have points over \(\mathbf{R}\)? I think so. If I understand Shimura’s paper here, I think the answer is yes.
3. Does this Shimura curve have points over \(\mathbf{Q}\)? I think so! Assuming it has points over \(\mathbf{R}\) you only need to check all other finite primes, and the one that is most worrying is \(p=7\) but you don’t really even need to check that one either if the others all work.
4. Assuming it is \(\mathbf{P}^1_{\mathbf{Q}}\), does that help at all? At the very least it provides succor that lots of \(A\) should exist over \(\mathbf{Q}\), but it’s not so clear how to go from a point to an equation. (Consider the easier case of Shimura curves corresponding to fake elliptic curves, for example.) Given a complex point, can one at least reconstruct some complex invariants of \(A\) such as its periods? Probably understanding this Shimura curve and its relationship with the moduli problem (over different fields) as concretely as possible would be a “second track” in this problem. (Presumably an advantage of a polymath project is that you can attack it from several angles at once.)

Thoughts welcome!