Let K be a number field and let g ≥ 2 be an integer. For a smooth projective geometrically connected curve C defined over K, write C(K) for the set of K-rational points of C, and #C(K) for its cardinality. Then there exists a number N(K,g), depending only on K and on g, such that #C(K) ;≤; N(K,g) for every smooth projective geometrically connected curve C defined over K whose genus equals g. Equivalently, the quantity sup_C ;#C(K), where the supremum is taken over all smooth projective geometrically connected curves C over K of genus g, is finite for each pair (K,g) with [K:Q]<∞ and g ≥ 2.

Uniform boundedness conjecture for rational points

Let $K$ be a number field and let $g \ge 2$ be an integer. For a smooth projective geometrically connected curve $C$ defined over $K$ , write $C(K)$ for the set of $K$ -rational points of $C$ , and $\#C(K)$ for its cardinality.

Then there exists a number $N(K,g)$ , depending only on $K$ and on $g$ , such that

\#C(K) \;\le\; N(K,g)

for every smooth projective geometrically connected curve $C$ defined over $K$ whose genus equals $g$ .

Equivalently, the quantity

\sup_{C} \;\#C(K),

where the supremum is taken over all smooth projective geometrically connected curves $C$ over $K$ of genus $g$ , is finite for each pair $(K,g)$ with $[K:\mathbb{Q}]<\infty$ and $g \ge 2$ .

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Equivalent formulations 2

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Uniform boundedness conjecture for rational points on curves
Let $C$ be a curve of genus $g>1$ defined over a number field $K$ .

Uniform boundedness conjecture. There exists a constant $B(g,K)$ such that
$\# C(K) < B(g,K).$
This conjecture asks for a bound on the number of $K$ -rational points depending only on the genus and the number field, not on the individual curve. It is a uniform strengthening of Faltings' theorem, which gives finiteness for each curve but no such uniform bound. The conjecture remains open in general.
source: Noam Kantor, “Rank-Favorable Bounds for Rational Points on Superelliptic Curves of Small Rank”, arXiv:1708.09120 (2017).
Faltings's uniform boundedness conjecture for rational points
Let $g\geq2$ be a genus and let $K$ be a number field. A smooth algebraic curve of genus $g$ defined over $K$ has a set of $K$ -rational points.

Uniform boundedness conjecture for rational points. There exists a constant $c>0$ , depending only on $g$ and $K$ , such that every smooth algebraic curve of genus $g$ defined over $K$ has at most $c$ rational points.

The source calls this a refinement of Faltings' theorem and uses it as an input for bounding rational realisations. The parser supplies no resolution status; the claim is recorded as open in the source context.
source: Sean Dewar, Nora Frankl, Samuel Mansfield, Anthony Nixon, Jonathan Passant and Audie Warren, “Generalised Erdős distance theory on graphs”, arXiv:2505.06590 (2025).

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Wikipedia, Uniform boundedness conjecture for rational points, the article this problem comes from.

algebra unsolved-problem

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Progress summary

Equivalent formulations 2

Uniform boundedness conjecture for rational points on curves

Faltings's uniform boundedness conjecture for rational points

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Additional references