Faltings height

Target entity: Faltings height
Context triple: [Faltings' theorem, relatedTo, Faltings height]

• A. Faltings' theorem
  Faltings' theorem is a landmark result in arithmetic geometry that proves every algebraic curve of genus greater than one over a number field has only finitely many rational points.
• B. Gross–Zagier formula
  The Gross–Zagier formula is a fundamental result in number theory that relates the heights of Heegner points on elliptic curves to the derivatives of associated L-functions, with deep implications for the Birch and Swinnerton-Dyer conjecture.
• C. Siegel's theorem on integral points
  Siegel's theorem on integral points is a fundamental result in number theory and Diophantine geometry stating that certain algebraic curves, notably those of genus at least one, have only finitely many integral points.
• D. Hasse–Weil bound for abelian varieties
  The Hasse–Weil bound for abelian varieties is a fundamental result in arithmetic geometry that gives sharp estimates for the number of rational points on abelian varieties over finite fields in terms of their dimension and the field size.
• E. Hasse bound for elliptic curves
  The Hasse bound for elliptic curves is a fundamental result in number theory that gives tight limits on how far the number of points on an elliptic curve over a finite field can deviate from the size of the field plus one.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Faltings height
Target entity description: The Faltings height is an arithmetic invariant that measures the complexity of an abelian variety (or algebraic curve) over number fields, playing a central role in Diophantine geometry and arithmetic geometry.

• A. Faltings' theorem
  Faltings' theorem is a landmark result in arithmetic geometry that proves every algebraic curve of genus greater than one over a number field has only finitely many rational points.
• B. Gross–Zagier formula
  The Gross–Zagier formula is a fundamental result in number theory that relates the heights of Heegner points on elliptic curves to the derivatives of associated L-functions, with deep implications for the Birch and Swinnerton-Dyer conjecture.
• C. Siegel's theorem on integral points
  Siegel's theorem on integral points is a fundamental result in number theory and Diophantine geometry stating that certain algebraic curves, notably those of genus at least one, have only finitely many integral points.
• D. Hasse–Weil bound for abelian varieties
  The Hasse–Weil bound for abelian varieties is a fundamental result in arithmetic geometry that gives sharp estimates for the number of rational points on abelian varieties over finite fields in terms of their dimension and the field size.
• E. Hasse bound for elliptic curves
  The Hasse bound for elliptic curves is a fundamental result in number theory that gives tight limits on how far the number of points on an elliptic curve over a finite field can deviate from the size of the field plus one.
• F. None of above. chosen