Modular Forms from Generalized Cohomology Theories

Definitions and mathematical statement

Coefficient-image question. The source describes stabilizing classifying spaces of elliptic bundles associated with genus-one conformal field theories to obtain generalized cohomology theories $E$ equipped with natural maps $\Phi:E\to K[[q]][q^{-1}]$. Are the images of their coefficient rings modular forms? The quantifier concerns theories and maps from this specified construction, not arbitrary cohomology theories or arbitrary maps into a Laurent-series target. The elliptic-bundle category, classifying spaces, stabilization, multiplicative structure, and maps are not constructed in the supplied record.

In a conventional spectral interpretation, the coefficient ring is $E_* = \pi_*(E)$, and the map induces $\Phi_*:E_*\to K_*[[q]][q^{-1}]$, where $K$ denotes the intended periodic $K$-theory and $q$ is a formal variable. Here $[[q]][q^{-1}]$ allows Laurent series with only finitely many negative powers in each element. For complex periodic $K$-theory one writes $K_*\cong{\mathbb{Z}}[\beta,\beta^{-1}]$, $|\beta|=2$. After specifying the grading convention, an even-degree coefficient $a\in E_{2w}$ has an image of the form $\beta^w f_a(q)$. The usual modularity interpretation asks whether $f_a(q)$ is the Fourier expansion at the chosen cusp of a modular form of the corresponding weight, for a specified modular group and coefficient ring. This is a conditional model of the claim; the source does not itself fix complex versus another $K$-theory, the weight-degree convention, or the normalization.

A modular form of integer weight $w$ for a specified subgroup $\Gamma$ of $\mathrm{SL}_2({\mathbb{Z}})$ is a holomorphic function on the upper half-plane satisfying $f((a\tau+b)/(c\tau+d))=(c\tau+d)^w f(\tau)$ for $\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\in\Gamma$, with the prescribed behavior at cusps and any specified multiplier. Its Fourier expansion uses a cusp parameter, conventionally $q=e^{2\pi i\tau}$ at a width-one cusp. Allowing poles at cusps gives a different, weakly holomorphic variant. Since the target allows negative powers of $q$, the source must specify which cusp condition it intends; a Laurent target alone does not enforce holomorphic modular forms. The intended claim is image containment in the relevant Fourier-expansion ring, not surjectivity onto all modular forms.

The supplied triage checked 2026-08-21 labels the item partially solved. It reports that elliptic spectra and topological modular forms realize major portions of the proposed picture, while the particular generalized-cohomology construction and coefficient-image claim are not a single fully settled theorem. No identification of this $E$ with topological modular forms, or transfer of a theorem to every proposed construction, is inferred. This dated assessment requires independent verification.

Short English statement

For the cohomology theories arising from the specified elliptic-bundle construction, ask whether their coefficient images are Fourier expansions of modular forms. The construction, grading, modular group, and cusp conditions are missing; the question asks for containment, not recovery of every modular form.

Sources

• Local UnsolvedMath registry, permanent ID 30000243, code catalogue entry; Modular Forms from Generalized Cohomology Theories, Oberwolfach Reports Open Problems.
• Geometric Topology and Connections with Quantum Field Theory (2006), Oberwolfach Reports, DOI 10.4171/owr/2005/27, pp. 1505--1546. https://doi.org/10.4171/owr/2005/27. Proposed year and year tag: 2005; no proposer named.
• Supplied literature label Kriz--Sati generalized cohomology: https://arxiv.org/abs/hep-th/0410293.
• Supplied literature label Elliptic cohomology overview: https://ncatlab.org/ncatlab/show/elliptic%20cohomology. The supplied URL contains a TeX escape before the percent sign; the percent-encoded URL is rendered here.
• Supplied literature label OWR report: https://doi.org/10.4171/owr/2005/27.
• Literature triage checked 2026-08-21, status partially_solved; record updated 2026-09-16. The corrected_verified field reports primary-report inspection, not independently checked here.