Mazur's torsion theorem

Target entity: Mazur's torsion theorem
Context triple: [Barry Mazur, knownFor, Mazur's torsion theorem]

• A. Mazur control theorem
  The Mazur control theorem is a fundamental result in Iwasawa theory that relates Selmer groups over infinite p-adic extensions to those over finite layers, allowing arithmetic information to be “controlled” across the tower.
• B. Shafarevich group of a torus
  The Shafarevich group of a torus is an arithmetic invariant measuring the failure of local-global principles for principal homogeneous spaces under an algebraic torus over a global field.
• C. Mordell–Weil theorem
  The Mordell–Weil theorem is a fundamental result in number theory stating that the group of rational points on an abelian variety (in particular, an elliptic curve) over a number field is finitely generated.
• D. Fontaine–Mazur conjecture
  The Fontaine–Mazur conjecture is a central open problem in number theory that predicts which p-adic Galois representations of number fields arise from geometry or from automorphic forms.
• E. 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.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Mazur's torsion theorem
Target entity description: Mazur's torsion theorem is a landmark result in number theory that classifies all possible torsion subgroups of elliptic curves over the rational numbers.

• A. Mazur control theorem
  The Mazur control theorem is a fundamental result in Iwasawa theory that relates Selmer groups over infinite p-adic extensions to those over finite layers, allowing arithmetic information to be “controlled” across the tower.
• B. Shafarevich group of a torus
  The Shafarevich group of a torus is an arithmetic invariant measuring the failure of local-global principles for principal homogeneous spaces under an algebraic torus over a global field.
• C. Mordell–Weil theorem
  The Mordell–Weil theorem is a fundamental result in number theory stating that the group of rational points on an abelian variety (in particular, an elliptic curve) over a number field is finitely generated.
• D. Fontaine–Mazur conjecture
  The Fontaine–Mazur conjecture is a central open problem in number theory that predicts which p-adic Galois representations of number fields arise from geometry or from automorphic forms.
• E. 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.
• F. None of above. chosen