Elliptic Curve Values of Rigid Cocycles

Definitions and mathematical statement

For an elliptic curve $E$ defined over a field contained in $L$, write $E(L)$ for its group of $L$-rational points. Let $f$ be a weight-two newform equipped with an attached elliptic rigid cocycle, in the sense intended by the supplied source. Retain the source notation $E$ for the elliptic curve involved, $J_f^+$ and $J_f^-$ for the two indicated cocycle components, and $J_f^\pm[\tau]$ for their values at a real-multiplication point $\tau$.

Conjectural statement, subject to unresolved source specifications. For every such attached elliptic rigid cocycle and every real-multiplication point $\tau$ in its intended evaluation domain, both values are rational over the field denoted by $H_\tau$: \[ J_f^+[\tau]\in E(H_\tau) \qquad\text{and}\qquad J_f^-[\tau]\in E(H_\tau). \] The supplied record does not define $H_\tau$, specify the ambient domain of $\tau$, give the construction or normalization of these evaluations, or state the hypotheses relating $f$, the cocycle, and $E$. These are unresolved qualifications: the displayed assertion records the intended field-of-definition claim but is not a self-contained formulation until those data are supplied.

Short English statement

The source conjectures that both indicated values of an elliptic rigid cocycle attached to a weight-two newform, at every admissible real-multiplication point $\tau$, lie in $E(H_\tau)$. The evaluation domain, field $H_\tau$, and required attachment hypotheses remain unspecified.

Sources

• Local registry: supplied UnsolvedMath source record, permanent ID $30004347$, Oberwolfach Reports Open Problems collection, exact source code catalogue entry, title "Elliptic Curve Values of Rigid Cocycles." The record lists proposed year 2019 and names no proposer.
• Supplied primary report citation: Modular Forms (2020), Oberwolfach Reports, pp. 3529--3606, DOI 10.4171/owr/2019/57. https://doi.org/10.4171/owr/2019/57. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions; no specific internal statement location is supplied.
• Supplied literature reference: Darmon--Vonk, "arithmetic intersections of modular geodesics." https://doi.org/10.1016/j.jnt.2020.12.012.
• Supplied literature reference: Darmon--Pozzi--Vonk, "special RM values." https://doi.org/10.4171/JEMS/1344.
• Supplied literature assessment, checked 2026-08-22: the general field-of-definition conjecture for real-multiplication values of the $f$-components of rigid meromorphic cocycles is labelled open, with current theorems described as covering special cocycles such as the Dedekind--Rademacher class. This is supplied dated curation derived from the Oberwolfach report collection, not an independently verified literature assessment; its status and cited claims require independent verification. The report above is also included among the supplied literature sources.