Vojta’s conjectures

Target entity: Vojta’s conjectures
Context triple: [abc conjecture, relatedTo, Vojta’s conjectures]

• A. Shafarevich conjecture for abelian varieties
  The Shafarevich conjecture for abelian varieties is a finiteness statement predicting that, over a number field, there are only finitely many isomorphism classes of abelian varieties with good reduction outside a fixed finite set of places, a result ultimately proved by Faltings.
• B. Weil conjectures
  The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
• C. Standard Conjectures on Algebraic Cycles
  The Standard Conjectures on Algebraic Cycles are a set of deep, still unproven hypotheses in algebraic geometry that aim to provide a foundational theory of algebraic cycles and their cohomological properties, underpinning much of the modern theory of motives.
• D. Tate Conjecture
  The Tate Conjecture is a major open problem in arithmetic geometry that predicts a deep connection between algebraic cycles on varieties over finite fields and their Galois-invariant étale cohomology classes.
• E. Bombieri–Lang conjecture
  The Bombieri–Lang conjecture is a major unsolved conjecture in number theory and arithmetic geometry predicting that varieties of general type over number fields have only finitely many rational points.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Vojta’s conjectures
Target entity description: Vojta’s conjectures are a collection of deep conjectures in Diophantine geometry that generalize and unify results in number theory and Nevanlinna theory, with far-reaching implications including those related to the abc conjecture.

• A. Shafarevich conjecture for abelian varieties
  The Shafarevich conjecture for abelian varieties is a finiteness statement predicting that, over a number field, there are only finitely many isomorphism classes of abelian varieties with good reduction outside a fixed finite set of places, a result ultimately proved by Faltings.
• B. Weil conjectures
  The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
• C. Standard Conjectures on Algebraic Cycles
  The Standard Conjectures on Algebraic Cycles are a set of deep, still unproven hypotheses in algebraic geometry that aim to provide a foundational theory of algebraic cycles and their cohomological properties, underpinning much of the modern theory of motives.
• D. Tate Conjecture
  The Tate Conjecture is a major open problem in arithmetic geometry that predicts a deep connection between algebraic cycles on varieties over finite fields and their Galois-invariant étale cohomology classes.
• E. Bombieri–Lang conjecture
  The Bombieri–Lang conjecture is a major unsolved conjecture in number theory and arithmetic geometry predicting that varieties of general type over number fields have only finitely many rational points.
• F. None of above. chosen