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.
be a non-singular algebraic variety, let
Choose Weil height functions
, a local height function
Fix a finite set of absolute values
and a non-empty Zariski open set
, depending on all of the above choices, such that Examples: There are generalizations in which
, 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.