Nontriviality of Heegner-Point Kolyvagin Systems

Definitions and mathematical statement

Let $E/{\mathbb{Q}}$ be an elliptic curve, let $p$ be an odd prime at which $E$ has good ordinary reduction, and let $K$ be an imaginary quadratic field, meaning a degree-two extension of ${\mathbb{Q}}$ with no real embedding. The supplied statement additionally requires $K$ to be admissible, but does not define this condition.

Write $\mathcal N$ for the intended indexing collection of square-free products of admissible rational primes inert in $K$. Here square-free means that no prime factor is repeated, and a rational prime $\ell$ is inert in $K$ if $\ell\mathcal O_K$ remains a prime ideal, where $\mathcal O_K$ is the ring of integers of $K$. The admissibility conditions on these primes and the convention concerning the empty product $1$ are not supplied.

Let $\{\kappa_n\}_{n\in\mathcal N}$ denote the Kolyvagin system derived from Heegner points referred to in the record. Its construction, coefficient modules, and any auxiliary choices are unspecified. Subject to the intended admissibility conditions and construction, the supplied target is the universal nonvanishing assertion \[ \text{for every such }(E,p,K),\qquad \exists n\in\mathcal N\quad \kappa_n\ne 0. \] Equivalently, the intended system is not identically zero; each comparison with zero is in the intended coefficient module of $\kappa_n$.

This is a schematic transcription of the supplied clean statement, not a complete independently determined conjecture: the missing admissibility conditions and coefficient data prevent a precise specification of its domain and classes. Moreover, the supplied original passage, "The first major progress towards this conjecture is due to W.", identifies neither the conjecture nor the person denoted by $W$. It does not by itself establish the connection to the Heegner-point formulation.

Short English statement

The supplied question asks whether every intended admissible elliptic-curve, prime, and imaginary-quadratic-field setup yields at least one nonzero Heegner-point Kolyvagin class. The admissibility conditions and the classes' coefficient modules are missing, and the original fragment does not identify this conjecture. The formulation therefore requires human mathematical and source review.

Sources

• Supplied local registry record: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30005573$, exact source problem code catalogue entry; title: Nontriviality of Heegner-Point Kolyvagin Systems. Collection: Oberwolfach Reports Open Problems. The record gives the proposed year as 2023 and supplies no proposer.
• Supplied report citation: Algebraische Zahlentheorie (2023), Oberwolfach Reports, DOI 10.4171/owr/2023/28, pp. 1543--1606. Supplied source and literature reference, labelled "Oberwolfach Report 28/2023": https://doi.org/10.4171/owr/2023/28. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature assessment dated 2026-08-22: the fragment about progress "due to W." does not identify a conjecture or complete claim, so no determinate solved or disproved status can be assigned. This is supplied dated curation, not independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the problem "open" and the statement corrected_verified; its verification note claims a check against the official report preserving hypotheses, notation, and target. These are supplied registry assertions, not verification performed for this draft. They do not resolve the missing formulation data or the disconnect between the original fragment and the clean statement.