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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 Erdos-Ulam problem. 1
Classification
Resolution Status
Open 1
Prize Status
Prize Status (category)
No Prize Offered 1
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
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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
View the Source
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