Unbounded Degrees of Full Belyi Maps

Definitions and mathematical statement

A Belyi map over ${\mathbb{Q}}$ is a finite nonconstant morphism $f:X\to{\mathbb{P}}^1_{{\mathbb{Q}}}$, where $X$ is a smooth, projective, geometrically connected curve over ${\mathbb{Q}}$, whose geometric branch locus is contained in $\{0,1,\infty\}$. Its degree is $\deg(f)=[{\mathbb{Q}}(X):{\mathbb{Q}}(t)]$, where $t$ is the standard coordinate on the target.

For a finite group $G$, put \[ \pi(G)=\{p\text{ prime}:p\mid |G|\}. \] In the supplied formulation, a finite set $P$ of rational primes is called anabelian if there exists a nonabelian finite simple group $G$ such that $\pi(G)\subseteq P$.

The source record uses the terms full Belyi map and bad reduction without supplying their definitions. Write $\operatorname{Full}(f)$ for the intended fullness condition and $\operatorname{Bad}(f)$ for the intended set of bad-reduction primes. These are placeholders for source-specific definitions, not definitions established by this draft. The record also does not specify whether additional restrictions on the source curve $X$ are implicit in the word "full." Subject to these unresolved qualifications, define \[ N(P)=\left\{n\in{\mathbb{Z}}:\ n\geq4,\ \exists\text{ a Belyi map }f\text{ over }{\mathbb{Q}}\text{ with }\deg(f)=n,\ \operatorname{Full}(f),\ \operatorname{Bad}(f)\subseteq P\right\}. \] Thus bad reduction is permitted only at primes in $P$; it is not required at every prime in $P$.

Conjecture (supplied formulation). For every finite set $P$ of rational primes, \[ \left(\exists\,G\text{ a nonabelian finite simple group with }\pi(G)\subseteq P\right) \quad\Longrightarrow\quad |N(P)|=\infty. \] Equivalently, for each such $P$ and every integer $B\geq4$, there should exist an integer $n>B$ belonging to $N(P)$. A fully specified mathematical statement requires recovery of the intended fullness and bad-reduction definitions, including any restrictions on $X$.

Short English statement

The supplied conjecture asks whether every finite prime set containing all prime divisors of the order of some nonabelian finite simple group supports full Belyi maps over ${\mathbb{Q}}$ of arbitrarily large degree, with bad reduction confined to that set. The meanings of "full" and "bad reduction," and any implicit restrictions on the source curve, must be recovered before this becomes a fully specified statement.

Sources

• Local UnsolvedMath registry: supplied complete record with permanent ID $30002917$, title "Unbounded Degrees of Full Belyi Maps," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2015 and names no individual proposer.
• Supplied report citation: Explicit Methods in Number Theory (2016), Oberwolfach Reports, DOI 10.4171/owr/2015/32, pp. 1809--1866; https://doi.org/10.4171/owr/2015/32. This is also the supplied literature source labelled "original Oberwolfach report." The background attributes the problem to workshop open-problem discussions documented in Oberwolfach Reports.
• Supplied literature source labelled "Roberts primary paper": https://www.numdam.org/item/10.5802/pmb.21.pdf. No further bibliographic details are supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that the unboundedness conjecture for Belyi maps with prescribed anabelian bad-reduction primes remains presented as a conjecture, with no proof located. It lists the report and Roberts paper above as checked sources. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and reports that the official report passage was checked directly and its question, definitions, and notation restored without strengthening the claim. These are supplied verification claims; this draft does not independently verify them or claim to have visited either source.