Math 2.0

Algebraic geometry Arithmetic and Diophantine geometry Conjecture

Lang's conjectures

Builds on Algebraic number fieldRational pointKodaira dimensionZariski topology

Statement

Let KK be an algebraic number field and let XX be a projective variety of general type over KK. This means that the Kodaira dimension of XX equals its dimension.

  1. The set X(K)X(K) of KK-rational points is not dense in XX for the Zariski topology.
  2. There is a nonempty Zariski open subset U⊆XU \subseteq X such that U(L)U(L) is finite for every finite extension LL of KK.

Remarks

Part 1 is often called the Bombieri–Lang conjecture. Bombieri posed it for surfaces in a 1980 lecture, and Lang posed it in general. For curves, general type means genus at least 22, and both parts follow from Faltings's theorem. The conjecture is open for surfaces. It follows from Vojta's conjecture. Caporaso, Harris and Mazur showed in 1997 that it implies a uniform bound for curves. The number of KK-rational points on a curve of genus g≥2g \ge 2 over KK would be bounded in terms of gg and KK only.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Algebraic number field, concept, no evidence levelAlgebraic number fieldconceptKodaira dimension, concept, no evidence levelKodaira dimensionconceptRational point, concept, no evidence levelRational pointconceptZariski topology, concept, no evidence levelZariski topologyconceptLang's conjectures, conjecture, no evidence levelLang's conjecturesconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Algebraic number field concept
  2. Kodaira dimension concept
  3. Rational point concept
  4. Zariski topology concept
  5. Lang's conjectures conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Bombieri%E2%80%93Lang_conjecture
About this page
TagTQ1QDAK2BR
URIm2://TQ1QDAK2BR
Kindconjecture
WritingW0 stub
MSC14G05, 11G35, 14G40
StatusThis copy of the site has no build data. Run m2 status TQ1QDAK2BR.
AboutRational point
UsesAlgebraic number field, Rational point, Kodaira dimension, Zariski topology
Changed2 November 2026

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