Geometric Interpretations of Lattice Modular Forms

Definitions and mathematical statement

Let $L_{K3}$ denote the even unimodular lattice of signature $(3,19)$. Let $M\subset L_{K3}$ be a primitive $2$-elementary Lorentzian sublattice. Here primitive means that $L_{K3}/M$ is torsion-free; Lorentzian means that the bilinear form on $M$ has signature $(1,r-1)$, where $r=\operatorname{rank}(M)$; and $2$-elementary means that $M^\vee/M\cong({\mathbb{Z}}/2{\mathbb{Z}})^a$ for some integer $a\geq0$, where \[ M^\vee=\{x\in M\otimes_{{\mathbb{Z}}}{\mathbb{Q}}:(x,m)\in{\mathbb{Z}}\text{ for every }m\in M\}. \] Put \[ \Lambda=M^\perp=\{x\in L_{K3}:(x,m)=0\text{ for every }m\in M\}. \] Write $F_\Lambda$ for the vector-valued modular form associated with $\Lambda$ in the cited report. The supplied record does not specify its defining formula, weight, representation, or normalization, so this notation remains dependent on the cited construction.

Question. For every primitive $2$-elementary Lorentzian sublattice $M\subset L_{K3}$, give a geometric interpretation of the associated form $F_{M^\perp}$. In particular, can the reported realizations of $F_\Lambda$ as the elliptic genus of a bundle over a K3 surface be extended beyond the cases in which such a realization is already known?

The supplied record does not identify those known cases or specify the category of bundles, the convention for the elliptic genus, or how that genus is identified with the vector-valued form. Consequently, a precise universal existence assertion for bundles cannot be extracted without further source information.

The earlier extracted wording also asks about $2^{g-1}F_\Lambda+f_\Lambda$. Neither $g$ nor $f_\Lambda$ is defined in the supplied record. Its verification note states that this alternative was omitted from the corrected formulation because it was not posed as a separate problem. It is retained here as a source qualification, without asserting a separate conjecture about that expression.

Short English statement

For each lattice obtained as the orthogonal complement of a primitive $2$-elementary Lorentzian sublattice of the K3 lattice, explain the geometry encoded by its associated modular form $F_\Lambda$. The particular aim is to extend reported interpretations as elliptic genera of bundles over K3 surfaces. Missing definitions and conventions prevent a fully specified bundle-existence statement.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003142$, title Geometric Interpretations of Lattice Modular Forms, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the record attributes the problem to workshop discussions documented in Oberwolfach Reports and gives proposed year 2016.
• Supplied report citation: Moduli spaces and Modular forms (2017), Oberwolfach Reports, pp. 1259--1318, DOI 10.4171/owr/2016/23. Source URL and supplied literature-source link labeled "original OWR report": https://doi.org/10.4171/owr/2016/23. The supplied title year is retained as given.
• The registry labels the statement corrected_verified. Its supplied verification note reports that the official report was checked, that the K3-lattice origin of $\Lambda$ was restored, and that an extracted alternative not posed as a separate problem was omitted. These are supplied verification claims, not independent verification performed for this draft.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that no authoritative general resolution was located for the extension to all $F_\Lambda$ or the indicated combinations. It lists the original OWR report above as the source checked. This is supplied dated curation, not an independently verified assessment; time-sensitive status and cited claims require independent checking.