On heights of points on algebraic varieties over number fields
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 conjecture.
Let formula_1 be a number field, let formula_2 be a non-singular algebraic variety, let formula_3 be an effective divisor on formula_4 with at worst normal crossings, let formula_5 be an ample divisor on formula_4, and let formula_7 be a canonical divisor on formula_4. Choose Weil height functions formula_9 and formula_10 and, for each absolute value formula_11 on formula_1, a local height function formula_13. Fix a finite set of absolute values formula_14 of formula_1, and let formula_16. Then there is a constant formula_17 and a non-empty Zariski open set formula_18, depending on all of the above choices, such that
formula_19
Examples: 
Generalizations.
There are generalizations in which formula_34 is allowed to vary over formula_35, and there is an additional term in the upper bound that depends on the discriminant of the field extension formula_36.
There are generalizations in which the non-archimedean local heights formula_13 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.