Uniform boundedness conjecture for rational points

In arithmetic geometry, the uniform boundedness conjecture for rational points asserts that for a given number field ${\displaystyle K}$ and a positive integer ${\displaystyle g\geq 2}$, there exists a number ${\displaystyle N(K,g)}$ depending only on ${\displaystyle K}$ and ${\displaystyle g}$ such that for any algebraic curve ${\displaystyle C}$ defined over ${\displaystyle K}$ having genus equal to ${\displaystyle g}$ has at most ${\displaystyle N(K,g)}$ ${\displaystyle K}$-rational points. This is a refinement of Faltings's theorem, which asserts that the set of ${\displaystyle K}$-rational points ${\displaystyle C(K)}$ is necessarily finite.