Uniform Primary Brauer Bounds for K3 Surfaces

Definitions and mathematical statement

A number field is a finite extension of ${\mathbb{Q}}$. A K3 surface over a number field $K$ means a smooth, projective, geometrically integral surface $X/K$ with $\omega_X\cong\mathcal O_X$ and $H^1(X,\mathcal O_X)=0$. For an abelian group $G$ and a prime $p$, define its $p$-primary torsion subgroup by \[ G[p^\infty]=\bigcup_{n\geq 1}\ker(p^n:G\longrightarrow G). \] The supplied source uses $\operatorname{Br}'(X)$ without defining the prime. Retaining that notation, the requested assertion is \[ \forall K\text{ a number field}\;\forall p\text{ a prime}\;\exists C_{K,p}\in{\mathbb{Z}}_{\geq 1}\;\forall X/K\text{ a K3 surface},\qquad \left|\operatorname{Br}'(X)[p^\infty]\right|\leq C_{K,p}. \] Thus the bound must be independent of $X$; no lattice polarization is fixed in this question. The intended definition of $\operatorname{Br}'(X)$ must be confirmed before this becomes a fully specified mathematical statement. In particular, the supplied record does not specify whether the group is the cohomological Brauer group or a quotient involving constant classes. The displayed question preserves the source notation without selecting an interpretation.

Short English statement

For each number field $K$ and prime $p$, ask whether the sizes of the $p$-primary torsion groups $\operatorname{Br}'(X)[p^\infty]$ are uniformly bounded over all K3 surfaces over $K$. The meaning of the prime in $\operatorname{Br}'$ remains unresolved in the supplied evidence.

Sources

• Local UnsolvedMath registry, permanent ID $30005958$, title Uniform Primary Brauer Bounds for K3 Surfaces, source problem code catalogue entry; supplied complete source record in the Oberwolfach Reports Open Problems collection.
• Supplied report citation: Algebraic Geometry: Wall Crossing and Moduli Spaces, Varieties and Derived Categories (2025), Oberwolfach Reports, pp. 1787--1854, DOI 10.4171/owr/2024/32. Original OWR report: https://doi.org/10.4171/owr/2024/32. The registry separately records proposed year 2024.
• The original statement attributes the question to Emma Brakkee's gong talk Bounding Brauer groups using moduli spaces, joint work with D. Bragg and A. Várilly-Alvarado. No separate proposer is identified in the supplied proposer field.
• Supplied literature source: Brakkee--Bragg--Varilly-Alvarado research summary, https://indico.sns.it/event/98/contributions/794/.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem partially solved and reports uniform $p$-primary bounds for K3 surfaces with a fixed lattice polarization, while describing the question over all K3 surfaces over the number field as open. This is supplied dated curation, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and says that it was reconstructed and checked against the official Oberwolfach report. These are supplied verification claims; no independent source consultation is asserted here.