Integral Points over Almost All Primitive Number Fields

Definitions and mathematical statement

Use the usual convention that a hyperbolic curve over ${\mathbb{Q}}$ is a smooth, geometrically connected curve $X/{\mathbb{Q}}$ whose smooth projective completion $\overline X$ has genus $g$, with $r=\#(\overline X\setminus X)(\overline{{\mathbb{Q}}})$ satisfying $2g-2+r>0$. The supplied record does not explicitly specify this convention.

Fix the model over ${\mathbb{Z}}$ referred to in the source, and denote it by $\mathcal X$, with generic fiber identified with $X$. For a number field $K$, let $\mathcal O_K$ be its ring of integers and set \[ \mathcal X(\mathcal O_K)=\operatorname{Hom}_{{\mathbb{Z}}}(\operatorname{Spec}\mathcal O_K,\mathcal X),\qquad \mathcal X({\mathbb{Z}})=\operatorname{Hom}_{{\mathbb{Z}}}(\operatorname{Spec}{\mathbb{Z}},\mathcal X). \] The source's notation $X(\mathcal O_K)$ and $X({\mathbb{Z}})$ is interpreted using this fixed model. These sets can depend on the model; the record does not specify additional conditions on it or clarify the quantifier over models.

Under the usual field-theoretic convention, call a number field $K$ primitive if there is no field $F$ with ${\mathbb{Q}}\subsetneq F\subsetneq K$. For an integer $n\geq1$ and a real number $B\geq1$, define \[ \mathcal P_n(B)=\left\{[K]:[K:{\mathbb{Q}}]=n,\ K\text{ is primitive},\ |\operatorname{Disc}(K/{\mathbb{Q}})|\leq B\right\}, \] where $[K]$ denotes a ${\mathbb{Q}}$-isomorphism class and the discriminant is the field discriminant. Thus ordering by discriminant is interpreted as counting isomorphism classes by absolute field discriminant.

Conjecture (supplied formulation). For every hyperbolic curve $X/{\mathbb{Q}}$ equipped with the intended fixed model $\mathcal X/{\mathbb{Z}}$, and every integer $n\geq1$, \[ \lim_{B\to\infty} \frac{\#\left\{[K]\in\mathcal P_n(B): \mathcal X(\mathcal O_K)= \operatorname{im}\left(\mathcal X({\mathbb{Z}})\longrightarrow\mathcal X(\mathcal O_K)\right) \right\}} {\#\mathcal P_n(B)}=1. \] The map in this formula is induced by ${\mathbb{Z}}\hookrightarrow\mathcal O_K$; equality means that every integral point over $\mathcal O_K$ comes from an integral point over ${\mathbb{Z}}$. Ratios are taken when the denominator is nonzero. The case $n=1$ is included: its only field is ${\mathbb{Q}}$, whose ring of integers is ${\mathbb{Z}}$, so the ratio equals $1$ for $B\geq1$. The model qualification remains unresolved and must be checked against the source before treating this as a definitive formulation.

Short English statement

For a hyperbolic curve over ${\mathbb{Q}}$ with its intended fixed integral model, the conjecture predicts that, for each degree $n\geq1$, a proportion tending to $100\%$ of primitive number fields, counted by absolute discriminant, introduce no integral points beyond those over ${\mathbb{Z}}$. The precise scope of the allowed integral models requires clarification.

Sources

• Local UnsolvedMath registry, permanent ID $30004748$, title: Integral Points over Almost All Primitive Number Fields; supplied source code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is $2021$.
• Supplied source citation: Explicit Methods in Number Theory (2022), Oberwolfach Reports, pp. 1805--1858, DOI 10.4171/owr/2021/34. https://doi.org/10.4171/owr/2021/34. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled "Oberwolfach report": https://ems.press/content/serial-article-files/46913.
• Supplied literature source, labeled "Khawaja--Siksek": https://arxiv.org/abs/2511.15635. No title or precise result is supplied.
• Supplied dated literature curation, checked 2026-08-22: status labeled open; the assessment reports that no proof of the stated integral-point equality for one hundred percent of primitive number fields was found, and that recent work proves only special cases of related new-point conjectures. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The supplied background expressly requests independent verification of time-sensitive status and cited claims.