Vojta's conjecture

Summary

In mathematics, Vojta's conjecture is a conjecture introduced by Paul Vojta (1987) about heights of points on algebraic varieties over number fields. The conjecture was motivated by an analogy between diophantine approximation and Nevanlinna theory (value distribution theory) in complex analysis. It implies many other conjectures in Diophantine approximation, Diophantine equations, arithmetic geometry, and mathematical logic.

Statement of the conjectureEdit

Let   be a number field, let   be a non-singular algebraic variety, let   be an effective divisor on   with at worst normal crossings, let   be an ample divisor on  , and let   be a canonical divisor on  . Choose Weil height functions   and   and, for each absolute value   on  , a local height function  . Fix a finite set of absolute values   of  , and let  . Then there is a constant   and a non-empty Zariski open set  , depending on all of the above choices, such that

 

Examples:

  1. Let  . Then  , so Vojta's conjecture reads   for all  .
  2. Let   be a variety with trivial canonical bundle, for example, an abelian variety, a K3 surface or a Calabi-Yau variety. Vojta's conjecture predicts that if   is an effective ample normal crossings divisor, then the  -integral points on the affine variety   are not Zariski dense. For abelian varieties, this was conjectured by Lang and proven by Faltings (1991).
  3. Let   be a variety of general type, i.e.,   is ample on some non-empty Zariski open subset of  . Then taking  , Vojta's conjecture predicts that   is not Zariski dense in  . This last statement for varieties of general type is the Bombieri–Lang conjecture.

GeneralizationsEdit

There are generalizations in which   is allowed to vary over  , and there is an additional term in the upper bound that depends on the discriminant of the field extension  .

There are generalizations in which the non-archimedean local heights   are replaced by truncated local heights, which are local heights in which multiplicities are ignored. These versions of Vojta's conjecture provide natural higher-dimensional analogues of the ABC conjecture.

ReferencesEdit

  • Vojta, Paul (1987). Diophantine approximations and value distribution theory. Lecture Notes in Mathematics. Vol. 1239. Berlin, New York: Springer-Verlag. doi:10.1007/BFb0072989. ISBN 978-3-540-17551-3. MR 0883451. Zbl 0609.14011.
  • Faltings, Gerd (1991). "Diophantine approximation on abelian varieties". Annals of Mathematics. 123 (3): 549–576. doi:10.2307/2944319. MR 1109353.