modularity theorem for elliptic curves over Q

Target entity: modularity theorem for elliptic curves over Q
Context triple: [Taniyama–Shimura–Weil conjecture, alsoKnownAs, modularity theorem for elliptic curves over Q]

• A. Modular curves and the Eisenstein ideal
  "Modular curves and the Eisenstein ideal" is a landmark 1977 paper by Barry Mazur that uses the arithmetic of modular curves and the structure of the Eisenstein ideal in Hecke algebras to prove deep results about rational torsion points on elliptic curves over the rational numbers.
• B. Ribet's theorem
  Ribet's theorem is a result in number theory that linked certain modular forms to Galois representations and played a crucial role in the proof of Fermat's Last Theorem.
• C. Serre’s conjecture on Galois representations
  Serre’s conjecture on Galois representations is a landmark statement in number theory that predicts which two-dimensional mod p Galois representations of the absolute Galois group of the rationals arise from modular forms.
• D. Eichler–Shimura theory
  Eichler–Shimura theory is a foundational framework in number theory and arithmetic geometry that connects modular forms with the cohomology of modular curves and the theory of elliptic curves.
• E. Mazur's deformation theory of Galois representations
  Mazur's deformation theory of Galois representations is a foundational framework in number theory that systematically studies how p-adic Galois representations can be deformed, with deep applications to modular forms and the proof of Fermat’s Last Theorem.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: modularity theorem for elliptic curves over Q
Target entity description: The modularity theorem for elliptic curves over Q is a landmark result in number theory stating that every elliptic curve defined over the rational numbers corresponds to a modular form, a fact central to the proof of Fermat’s Last Theorem.

• A. Modular curves and the Eisenstein ideal
  "Modular curves and the Eisenstein ideal" is a landmark 1977 paper by Barry Mazur that uses the arithmetic of modular curves and the structure of the Eisenstein ideal in Hecke algebras to prove deep results about rational torsion points on elliptic curves over the rational numbers.
• B. Ribet's theorem
  Ribet's theorem is a result in number theory that linked certain modular forms to Galois representations and played a crucial role in the proof of Fermat's Last Theorem.
• C. Serre’s conjecture on Galois representations
  Serre’s conjecture on Galois representations is a landmark statement in number theory that predicts which two-dimensional mod p Galois representations of the absolute Galois group of the rationals arise from modular forms.
• D. Eichler–Shimura theory
  Eichler–Shimura theory is a foundational framework in number theory and arithmetic geometry that connects modular forms with the cohomology of modular curves and the theory of elliptic curves.
• E. Mazur's deformation theory of Galois representations
  Mazur's deformation theory of Galois representations is a foundational framework in number theory that systematically studies how p-adic Galois representations can be deformed, with deep applications to modular forms and the proof of Fermat’s Last Theorem.
• F. None of above. chosen