Moduli of lattice-polarized K3 surfaces and boundedness of Brauer groups
Danny Bragg, Emma Brakkee, Anthony Várilly-Alvarado
TL;DR
The paper constructs moduli stacks of lattice-polarized K3 surfaces over $ ext{Spec}\mathbb{Z}$, introducing very small ample cones to obtain separated moduli away from the supersingular locus and extending the framework to lattice-polarized twisted K3 surfaces to track Brauer classes. It develops complex-period theory for both untwisted and twisted lattices, showing that coarse moduli spaces decompose into orthogonal modular varieties, and, in the rank-19 case, that the components are unions of curves with genus and gonality growing with Brauer level, implying finiteness of points over number fields of bounded degree for sufficiently high level. The work then leverages this moduli-theoretic structure, together with orthogonal group index bounds and Fuchsian-group theory, to prove uniform $ ext{ell}$-primary bounds for Brauer groups in 1-dimensional families, providing a moduli-driven proof of Cadoret–Charles and related uniform-boundedness results. Overall, the paper links lattice-polarized K3 moduli, period maps, and Brauer-group arithmetic, offering a robust platform for uniform-boundedness questions in arithmetic geometry of K3 surfaces.
Abstract
Inspired by constructions over the complex numbers of Dolgachev and Alexeev-Engel, we define moduli stacks $\mathcal{M}_{(L,\mathcal{A})/\mathbb{Z}}$ of lattice-polarized K3 surfaces over arbitrary bases, paying particular attention to the open locus $\mathcal{P}_{(L,\mathcal{A})/\mathbb{Z}}$ of primitive lattice polarizations. We introduce the notion of very small ample cones $\mathcal{a}$, after Alexeev and Engel's small cones, to construct smooth, separated stacks of lattice polarized K3 surfaces $\mathcal{P}_{(L,\mathcal{a})/\mathbb{Z}[1/N]}$ over suitable open subsets of $\textrm{Spec}(\mathbb{Z})$. We add level structures, coming from classes in $\mathrm{H}^2(X,μ_n)$, to build moduli stacks $\mathcal{P}^{[n]}_{(L,\mathcal{A})/\mathbb{Z}}$ with a natural action by $\mathcal{P}_{(L,\mathcal{A})}\otimes \mathbb{Z}/n\mathbb{Z}$ whose associated quotient $\mathcal{Q}^{[n]}_{(L,\mathcal{A})}$ contains an open substack $\mathcal{Q}^{(n)}_{(L,\mathcal{A})}$ whose points parametrize pairs K3 surfaces $X$ such that $\textrm{Pic}(X) \simeq L$, together with a class $α\in \textrm{Br}(X)$ of order $n$. When $L$ has rank 19, we show that the coarse moduli space $\mathrm{Q}_{(L,\mathcal{a}),\mathbb{C}}^{(n)}$ is a union of quasi-projective curves, each isomorphic to an open subvariety of the quotient of the upper half plane by a discrete subgroup of $\mathrm{SL}_2(\mathbb{R})$. Fixing a prime $\ell$, we use this comparison to prove that the genus and the gonality of the components of $\mathrm{Q}_{(L,\mathcal{a}),\mathbb{C}}^{(\ell^{m})}$ grows with $m$, and hence that they have finitely many points over number fields of bounded degree. As an application, we furnish a new proof of a result by Cadoret--Charles, showing uniform boundedness of the $\ell$-primary torsion of Brauer groups of K3 surfaces over number fields varying in a $1$-dimensional lattice-polarized family.