Algebraic Cycles Realizing Modular Differential Forms

Definitions and mathematical statement

For a variety $X$, the conventional notation $CH^c(X)$ denotes the group of codimension-$c$ algebraic cycles modulo rational equivalence. Tensoring with ${\mathbb{Q}}$ gives rational coefficients. The supplied record does not define $r,K,W_r,A,\psi$, the differential forms, or the correspondence action denoted by $\Pi^*_{\mathrm{dR}}$. Consequently, the following preserves the source conjecture for its intended data, whose hypotheses must be recovered before this becomes a fully specified mathematical statement.

The conjecture asks whether there exist \[ \Pi^*\in CH^{r+1}(W_r\times A^{r+1})(K)\otimes{\mathbb{Q}} \quad\text{and}\quad q\in{\mathbb{Q}}\setminus\{0\} \] such that \[ \Pi^*_{\mathrm{dR}}\left([\omega_A^{r+1}]\right) =q[\omega_\psi^{r+1}] \] and \[ \Pi^*_{\mathrm{dR}}\left([\omega_A^j\overline{\omega}_A^{r+1-j}]\right)=0 \qquad\text{for every integer }j\text{ with }1\leq j\leq r. \] Thus the source's symbol $\sim$ means equality up to a nonzero rational scalar. The source imposes the vanishing condition only for $1\leq j\leq r$; no vanishing condition at either endpoint $j=0$ or $j=r+1$ is added.

The notation $(K)$ on the Chow group, the allowed values of $r$, the cohomology groups containing the displayed classes, and the meaning of the powers and conjugation in these expressions are unspecified. These expressions are retained as source notation; no interpretation as particular tensor products, exterior products, or cohomological projections is assumed.

Short English statement

For the geometric and differential-form data intended by the source, the conjecture asks for an algebraic correspondence with rational coefficients that sends the displayed pure differential class to a nonzero rational multiple of the modular differential class and annihilates every displayed mixed class indexed by $1\leq j\leq r$. The supplied record omits the definitions and hypotheses needed to specify those data and the correspondence action.

Sources

• Local registry: supplied UnsolvedMath source record, permanent ID $30001302$, title Algebraic Cycles Realizing Modular Differential Forms, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2009; no individual proposer is specified.
• Supplied report citation: Explicit Methods in Number Theory (2010), Oberwolfach Reports, pp. 1843--1920, DOI 10.4171/owr/2009/33. https://doi.org/10.4171/owr/2009/33. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference, labelled "Darmon Oberwolfach report": https://ems.press/content/serial-article-files/46233?nt=1.
• Supplied literature reference, labelled "Generalized Heegner cycles theorem": https://doi.org/10.1215/00127094-2142056.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that the conjectural algebraic cycle and complex modular parametrisation remain conditional, while Bertolini--Darmon--Prasanna prove a substantial $p$-adic Abel--Jacobi and Rankin-series analogue. This is supplied dated curation derived from the OWR material, not an independently verified assessment. The source explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and describes the algebraic-cycle realization conjecture as complete with normalized cycle, de Rham, and conjugate-form notation. These are supplied registry qualifications, not independent mathematical verification of this draft.