Uniform Boundedness Conjecture for Rational Points

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A number field $K$ is a finite extension of $\mathbb Q$, with degree $d=[K:\mathbb Q]\ge1$. A curve here means a smooth projective geometrically connected one-dimensional variety $C$ over $K$. Its genus is $g=\dim_K H^0(C,\Omega_C^1)$, the dimension of its regular differential forms. The set $C(K)$ consists of its $K$-rational points.

Statement recorded in the supplied source.

For every pair of integers $g\ge2$ and $d\ge1$, does there exist an integer $N(g,d)$ such that, for every number field $K$ of degree $d$ and every such curve $C/K$ of genus $g$, \[ \#C(K)\le N(g,d)? \] The bound may depend only on $g$ and $d$, not on the particular field, curve, coefficients, or rank of its Jacobian. [S2]

Short English statement

Can the number of rational points on every smooth curve of genus at least two be bounded using only its genus and the degree of its number field?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1470 (ALG-015). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eric Katz, Joseph Rabinoff and David Zureick-Brown, Uniform Bounds for the Number of Rational Points on Curves of Small Mordell–Weil Rank, arXiv:1504.00694v3 (2017), abstract and Section 1, full uniform-bound conjecture preceding Theorem 1.1. https://arxiv.org/pdf/1504.00694v3.