Vojta's conjecture

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 theory, diophantine equations, arithmetic geometry, and logic.

Let ${\displaystyle F}$ be a number field, let ${\displaystyle X/F}$ be a non-singular algebraic variety, let ${\displaystyle D}$ be an effective divisor on ${\displaystyle X}$ with at worst normal crossings, let ${\displaystyle H}$ be an ample divisor on ${\displaystyle X}$, and let ${\displaystyle K_{X}}$ be a canonical divisor on ${\displaystyle X}$. Choose Weil height functions ${\displaystyle h_{H}}$ and ${\displaystyle h_{K_{X}}}$ and, for each absolute value ${\displaystyle v}$ on ${\displaystyle F}$, a local height function ${\displaystyle \lambda _{D,v}}$. Fix a finite set of absolute values ${\displaystyle S}$ of ${\displaystyle F}$, and let ${\displaystyle \epsilon >0}$. Then there is a constant ${\displaystyle C}$ and a non-empty Zariski open set ${\displaystyle U\subseteq X}$, depending on all of the above choices, such that

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