Galois Equivariance of Generalized Heegner-Cycle Images

Definitions and mathematical statement

Let $K^{\mathrm{ab}}$ denote the maximal abelian extension of the field $K$ in a fixed separable closure. The supplied record does not specify $K$, the admissible values of $r$, the variety $X_r$, the object $A$, the map $\Phi_C$, or the construction and hypotheses defining a generalized Heegner cycle $\Delta_\varphi$. It also does not define the subscript $0$ in $CH^{r+1}(X_r)_0(H)$. These symbols therefore retain their source-dependent meanings; the following is a conditional transcription requiring those definitions before it becomes a fully specified conjecture.

The notation $CH^{r+1}$ denotes a Chow group of codimension-$r+1$ cycles modulo rational equivalence. The record does not specify its coefficient convention or precisely how the notation ${}_0(H)$ is formed. In the asserted target, $A(H)$ denotes the group of $H$-rational points of the intended object $A$, and $A(H)\otimes_{{\mathbb{Z}}}{\mathbb{Q}}$ denotes its rationalization, provided the required commutative group structure has been specified.

Conjecture as supplied. For every field $H\subseteq K^{\mathrm{ab}}$ and every generalized Heegner cycle \[ \Delta_\varphi\in CH^{r+1}(X_r)_0(H), \] the source asks whether \[ \Phi_C(\Delta_\varphi)\in A(H)\otimes_{{\mathbb{Z}}}{\mathbb{Q}} \] and \[ \Phi_C(\Delta_\varphi^\sigma) =\Phi_C(\Delta_\varphi)^\sigma \qquad\text{for every }\sigma\in\operatorname{Gal}(H/K). \] For the displayed Galois group to have its stated meaning, $H$ must contain $K$; thus the intended field condition appears to be $K\subseteq H\subseteq K^{\mathrm{ab}}$. This containment is implicit rather than explicit in the supplied statement. No finiteness condition on $H/K$ is supplied. The constructions of both Galois actions and the identification needed to interpret the displayed membership in the target of $\Phi_C$ remain unspecified.

Short English statement

The conjecture asks whether the image of every generalized Heegner cycle defined over an abelian extension $H$ of $K$ is an $H$-rational point of the intended group $A$, with rational coefficients, and whether taking this image commutes with every element of $\operatorname{Gal}(H/K)$. The source omits definitions and hypotheses needed to make this a fully specified mathematical statement.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001303$, title: Galois Equivariance of Generalized Heegner-Cycle Images; source problem code: catalogue entry. Collection: Oberwolfach Reports Open Problems. The registry records proposed year 2009 and supplies no named proposer.
• Supplied source 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 source, labeled "Darmon Oberwolfach report": https://ems.press/content/serial-article-files/46233?nt=1. This label alone does not establish the proposer of this conjecture.
• Supplied literature source, labeled "Generalized Heegner cycles theorem": https://doi.org/10.1215/00127094-2142056. The record does not provide a theorem number or its hypotheses.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that the complex algebraicity conjecture is not known in full generality, while $p$-adic analogues establish predicted algebraicity properties in broad settings. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.
• The registry supplies the statement label corrected_verified and the verification description that the field-of-definition and Galois-equivariance conjecture is complete and its notation normalized. These are supplied registry assertions, not independent verification of this draft.