Vojta's conjecture
In mathematics, Vojta's conjecture is a conjecture introduced by 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 conjecture 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: Let . Then , so Vojta's conjecture reads for all . 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.