Uniform Lower Bounds for Specialized Canonical Heights

Definitions and mathematical statement

Let $K$ be a number field, and let $(f_t)_{t\in U}$ be the supplied non-isotrivial family of rational maps on ${\mathbb{P}}^1$, parametrized by a curve $U$, with generic degree $d\geq 2$. In the usual function-field interpretation, non-isotrivial means that the generic map is not conjugate, over an algebraic closure of $K(U)$, to a rational map defined over $\overline K$. This interpretation presupposes an integral base curve; the source does not specify further hypotheses on $U$.

For a rational map $g:{\mathbb{P}}^1\to{\mathbb{P}}^1$ of degree $e\geq2$, its canonical height is \[ \widehat h_g(z)=\lim_{n\to\infty}e^{-n}h_{{\mathbb{P}}^1}\left(g^{\circ n}(z)\right), \] where $h_{{\mathbb{P}}^1}$ is the absolute logarithmic Weil height. Write $h(t)$ for the parameter height appearing in the source. Its defining divisor or line bundle, normalization, and additive representative are not supplied; these are unresolved parts of the formulation.

Provided that $h$ has been specified and every $t\in U(K)$ gives a defined specialization $f_t$ of degree at least $2$, the supplied question asks whether there exist constants $\varepsilon>0$ and $N\in{\mathbb{Z}}_{\geq0}$, independent of $t$, such that \[ \forall t\in U(K),\qquad \#\left\{z\in{\mathbb{P}}^1(K):\widehat h_{f_t}(z)<\varepsilon h(t)\right\}\leq N. \] Thus the exceptional points may depend on $t$, but their number must be uniformly bounded. The constants are sought for the fixed number field, family, base curve, and parameter height; no uniformity across different such data is asserted.

The source assumes only generic degree $d\geq2$. It does not explicitly exclude parameters at which specialization is undefined or has degree less than $2$. If such parameters belong to $U(K)$, the displayed question is not defined there using the canonical-height definition above. No deletion of parameters, extension to degenerate fibers, or replacement of the exact lower bound by a bound with an additive error is asserted in this draft.

Short English statement

For a fixed non-isotrivial family over a number field, ask whether every specialized map has canonical height at least a fixed positive multiple of the parameter height for all but a uniformly bounded number of $K$-rational points. The parameter height and the treatment of undefined or degree-less-than-$2$ specializations require clarification.

Sources

• Supplied local UnsolvedMath registry record, data/unsolvedmath/problems.json, permanent ID $30003121$, source problem code catalogue entry, title: Uniform Lower Bounds for Specialized Canonical Heights. Collection: Oberwolfach Reports Open Problems. No individual proposer is identified; the supplied proposed year is $2016$.
• Supplied report citation: Diophantische Approximationen (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/21, pp. 1099--1187. https://doi.org/10.4171/owr/2016/21. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The citation's year $2017$ and the $2016$ appearing in the DOI and proposed-year metadata are retained without reconciliation.
• Supplied literature source, labeled "OWR report": https://ems.press/content/serial-article-files/46625?nt=1.
• Supplied literature source, labeled "Ingram": https://academic.oup.com/imrn/article-abstract/2019/15/4859/4782491. No complete bibliographic citation or specific theorem is supplied for this item.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially_solved and states that the general uniform lower-bound conjecture remains open, while weighted-homogeneous and related special cases have been proved or obtained conditionally. This is supplied dated curation, not an independently verified assessment. The background expressly requests independent verification of time-sensitive status and cited claims; it does not identify the hypotheses or precise results for those special cases.
• The record labels its statement corrected_verified and says that mathematical content was preserved while notation, typography, or omitted definitions were repaired. This is a supplied editorial qualification, not mathematical verification performed for this draft.