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}
  
 that 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.
 The first significant progress towards the conjecture was due to Caporaso, Harris, and Mazur. They proved that the conjecture holds if one assumes the Bombieri–Lang conjecture.
 A variant of the conjecture, due to Mazur, asserts that there should be a number 
  
    
      
        N
        (
        K
        ,
        g
        ,
        r
        )
      
    
    {\displaystyle N(K,g,r)}
  
 such that for any algebraic curve 
  
    
      
        C
      
    
    {\displaystyle C}
  
 defined over 
  
    
      
        K
      
    
    {\displaystyle K}
  
 having genus 
  
    
      
        g
      
    
    {\displaystyle g}
  
 and whose Jacobian variety 
  
    
      
        
          J
          
            C
          
        
      
    
    {\displaystyle J_{C}}
  
 has Mordell–Weil rank over 
  
    
      
        K
      
    
    {\displaystyle K}
  
 equal to 
  
    
      
        r
      
    
    {\displaystyle r}
  
, the number of 
  
    
      
        K
      
    
    {\displaystyle K}
  
-rational points of 
  
    
      
        C
      
    
    {\displaystyle C}
  
 is at most 
  
    
      
        N
        (
        K
        ,
        g
        ,
        r
        )
      
    
    {\displaystyle N(K,g,r)}
  
. This variant of the conjecture is known as Mazur's Conjecture B.
 Michael Stoll proved that Mazur's Conjecture B holds for hyperelliptic curves with the additional hypothesis that 
  
    
      
        r
        ≤
        g
        −
        3
      
    
    {\displaystyle r\leq g-3}
  
. Stoll's result was further refined by Katz, Rabinoff, and Zureick-Brown in 2015. Both of these works rely on Chabauty's method.
 Mazur's Conjecture B was resolved by Dimitrov, Gao, and Habegger in a preprint in 2020 which has since appeared in the Annals of Mathematics using the earlier work of Gao and Habegger on the geometric Bogomolov conjecture instead of Chabauty's method.