In arithmetic geometry, the uniform boundedness conjecture asserts that for a given number field and positive integer, there exists a bound depending only on these parameters such that any algebraic curve defined over the number field with a specified genus has at most that many rational points. This refines Faltings theorem by establishing a uniform bound. It is also known as Mazur Conjecture B, one of several distinct conjectures proposed by Barry Mazur.
Facts
Partially Attested
Progress Toward ResolutionMazur's conjecture B, a weaker variant of the uniform boundedness conjecture, was resolved by Dimitrov, Gao, and Habegger in 2021. 1 A weaker variant known as Mazur's conjecture B was resolved in 2021; the full uniform boundedness conjecture for all genus remains open. StatementFor a given number field K and a positive integer g of at least 2, there exists a number N(K,g), depending only on K and g, such that any algebraic curve C defined over K having genus equal to g has at most N(K,g) K-rational points. 1 Classification
Resolution Status Prize Status
Prize Status (category) Sources
1. Uniform Boundedness Conjecture for Rational Points (Wikipedia)
Lead section
In arithmetic geometry, the uniform boundedness conjecture for rational points asserts that for a given number field K and a positive integer g >= 2, there exists a number N(K,g) depending only on K and g such that for any algebraic curve C defined over K having genus equal to g has at most N(K,g) K-rational points.
Progress section
Mazur's conjecture B was resolved by Dimitrov, Gao, and Habegger in 2021.
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