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