Mathematics Atlas

How Proof Is Made
Sign In
Text size
100%
Theme

Switch Atlas

Conjecture

Bombieri-Lang Conjecture

Algebraic Geometry

In arithmetic geometry, the Bombieri-Lang conjecture is an unsolved problem conjectured by Enrico Bombieri and Serge Lang about the Zariski density of the set of rational points of an algebraic variety of general type. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/

Facts
Statement
If X is a positive-dimensional algebraic variety of general type defined over a number field k, then the k-rational points of X do not form a dense set in the Zariski topology. 1
Proposed Year
1971 2
Proposed Year
1980 2
Progress Toward Resolution
Unsolved. In 1997 Caporaso, Mazur, Harris and Pacelli showed it implies a uniform boundedness conjecture for rational points, and it would resolve the Erdős-Ulam problem. 1
Classification
Resolution Status
Open 1
Prize Status
Prize Status (category)
No Prize Offered 1
Connections

In Branch

Algebraic Geometry, Branches of Mathematics
Source Bombieri-Lang conjecture (Wikipedia)

Posed By

Enrico Bombieri, Mathematicians
Source Bombieri-Lang conjecture (Wikipedia)
Serge Lang, Mathematicians
Source Bombieri-Lang conjecture (Wikipedia)
Sources
1. Bombieri-Lang Conjecture (Wikipedia)
Wikimedia Foundation
  • Lead section
    the Bombieri-Lang conjecture is an unsolved problem conjectured by Enrico Bombieri and Serge Lang about the Zariski density of the set of rational points of an algebraic variety of general type
  • Statement section
    if X is a positive-dimensional algebraic variety of general type defined over a number field k, then the k-rational points of X do not form a dense set in the Zariski topology.
  • Consequences section
    If true, the Bombieri-Lang conjecture would resolve the Erdős-Ulam problem
View the Source
2. Bombieri-Lang conjecture (Wikipedia)
  • History: "Independently in a series of papers starting in 1971
  • History: "In a 1980 lecture at the University of Chicago
  • In Branch: Algebraic Geometry, Lead
    the Bombieri-Lang conjecture is an unsolved problem conjectured by Enrico Bombieri and Serge Lang about the Zariski density of the set of rational points of an algebraic variety of general type
  • Posed By: Enrico Bombieri, Lead section
    In arithmetic geometry, the Bombieri-Lang conjecture is an unsolved problem conjectured by Enrico Bombieri and Serge Lang about the Zariski density of the set of rational points of an algebraic variety of general type.
  • Posed By: Serge Lang, Lead section
    In arithmetic geometry, the Bombieri-Lang conjecture is an unsolved problem conjectured by Enrico Bombieri and Serge Lang about the Zariski density of the set of rational points of an algebraic variety of general type.
View the Source
Comments (0)
No comments yet. Be the first to share a thought.
Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.