Vojta's conjecture, introduced by Paul Vojta, concerns heights of points on algebraic varieties over number fields. The conjecture was motivated by an analogy between Diophantine approximation and Nevanlinna theory in complex analysis, and it implies many other conjectures in Diophantine approximation, Diophantine equations, arithmetic geometry, and mathematical logic. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementA conjecture about heights of points on algebraic varieties over number fields, motivated by an analogy between diophantine approximation and Nevanlinna theory (value distribution theory) in complex analysis. 1 Proposed Year Progress Toward ResolutionOne predicted consequence, that S-integral points on the complement of an effective ample normal crossings divisor in a variety with trivial canonical bundle are not Zariski dense, was conjectured by Lang and proven by Faltings in the case of abelian varieties. 1 Classification
Resolution Status Prize Status
Prize Status (category) Sources
1. Vojta's Conjecture (Wikipedia)
Wikimedia FoundationLead section
the conjecture was introduced by Paul Vojta in 1987
Lead section, opening sentences
In mathematics, Vojta's conjecture is a conjecture introduced by Paul Vojta about heights of points on algebraic varieties over number fields.
Examples section
For abelian varieties, this was conjectured by Lang and proven by Faltings.
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