Potential Modularity of K3 surfaces

This post is to report on results of my student Chao Gu who is graduating this (academic) year.

If \(A/F\) is an abelian surface, then one can associate to \(A\) a K3 surface \(X\) (the Kummer surface) by blowing up \(A/[-1]\) at the \(16\) singular points (corresponding to \(2\)-torsion points of \(A\). If \(F\) is a totally real field, then one knows that \(A\) is (potentially) automorphic, and it follows that \(X\) is also (potentially) automorphic, which in particular implies the Hasse-Weil conjecture for \(X\). It also proves that

\( \rho(X/F) = – \mathrm{ord}_{s=1} L(H^2(X/\overline{F},\mathbf{Q}_p(1)),s),\)

where \(H^2(X/\overline{F},\mathbf{Q}_p(1))\) is the etale cohomology group considered as a Galois representation of \(G_F\); this was conjectured by Tate in the same paper where he makes the “usual” Tate conjecture on algebraic cycles. Not all K3 surfaces arise in this way. For a start, if \(A\) has (geometric) Picard rank \(\rho(A) \ge 1\), then \(X\) has geometric Picard rank \(16 + \rho(A) \ge 17\). If the Picard rank of \(X\) is at least \(19\), then \(X\) also has to arise (at least in the category of Motives) as a Kummer surface, but more subtly this is not true in rank \(17\) and \(18\), where there are further obstructions relating to the structure of the transcendental lattice (as first observed by Morrison in this paper). What Chao does is prove the following:

Theorem: [Chao Gu] Let \(X/\mathbf{Q}\) be a K3 surface of Picard rank at least \(17\). Then \(X\) is potentially automorphic, and the Hasse-Weil conjecture holds for \(X\).

In the most interesting case of rank \(17\), the approach is to lift the compatible family of \(5\)-dimensional orthogonal representations associated to the transcendental lattice to a compatible family of \(4\)-dimensional symplectic representations which one hopes to prove is potentially automorphic. Finding (motivic) lifts of K3 surfaces is a well-studied problem, and a nice analysis of what happens arithmetically can be found in Patrikis’ thesis. From the Kuga-Satake construction, one can certainly reduce to considering certain abelian varieties. The question is then narrowing down the precise endomorphism structures of these varieties as well as their fields of definition. It turns out that for the problem of interest, there are more or less three types of abelian varieties one might want to consider beyond abelian surfaces over a totally real field \(F\):

1. Abelian varieties \(A/F\) of dimension \(2d\), where \(A\) admits endomorphisms by an order in the ring of integers of a totally real field \(E\) of degree \([E:\mathbf{Q}] = d\).
2. Abelian surfaces \(A/H\) over some Galois extension \(H/F\) where the conjugate of \(A\) by \(\mathrm{Gal}(H/F)\) are all isogenous over \(H\).
3. Abelian \(4\)-folds \(A/F\) with endomorphisms by an order in a quaternion algebra \(D/\mathbf{Q}\).

More generally, one needs to consider the “cross-product” where several (or all) of these phenomena may occur at once. For those more familiar with the story of two-dimensional Galois representations over \(\mathbf{Q}\), these three extensions correspond to replacing elliptic curves over \(\mathbf{Q}\) by abelian varieties of \(\mathrm{GL}_2\)-type, to \(\mathbf{Q}\)-curves, and to fake elliptic curves respectively. It turns out that the last case doesn’t happen over totally real fields but the analog for abelian surfaces does, requiring certain conjectures to be modified.

The optimal generalization of Boxer-Calegari-Gee-Pilloni to this setting would be to prove that all of these varieties are potentially modular. However, it turns out that there is an obstruction to proving this: namely, is not always possible to prove that these varieties have enough ordinary primes (one needs something slightly stronger, namely ordinary primes whose unit eigenvalues are distinct modulo \(p\)). This puts some restrictions on what can be proved unconditionally, but everything works as long as there are enough ordinary primes. Chao’s proof requires a number of modifications from [BCGP], in particular to the Moret-Bailly part of the argument. In our original paper, we exploited the fact that we were working only with abelian surfaces which allowed us to use some tricks to simplify this step. In particular, the problem of finding an appropriate point on
the desired moduli space over \(\mathbf{Q}_p\) was made much simpler by virtue of the fact that the original abelian surface produced such a point. In Chao’s generalization, however,
this trick doesn’t work, and one must use more subtle arguments using Serre–Tate theory. Fortunately, enough tricks are available concerning ordinary primes to settle the general case of K3 surfaces of (geometric) Picard rank at least \(17\) when they are defined over \(\mathbf{Q}\). But note there do exist many such K3 surfaces (not related to abelian surfaces) over \(\mathbf{Q}\) that one can write down explicitly; see the examples due to Nori discussed in Section 9.4.

Note that this result is new even for Picard rank \(18\). For Picard rank \(19\) and \(20\), the (potential) modularity of any \(X/F\) for a totally real field \(F\) reduces to the corresponding problem for elliptic curves. The case of Picard rank \(16\) appears as hopeless as the case of generic genus three curves.