Eichler–Shimura theory

Target entity: Eichler–Shimura theory
Context triple: [Goro Shimura, knownFor, Eichler–Shimura theory]

• A. Shimura correspondence
  The Shimura correspondence is a fundamental result in number theory that establishes a deep link between modular forms of half-integral weight and modular forms of integral weight, with important applications to L-functions and arithmetic geometry.
• B. Shimura varieties
  Shimura varieties are higher-dimensional algebraic varieties that generalize modular curves and play a central role in the Langlands program by connecting number theory, automorphic forms, and arithmetic geometry.
• C. Shimura reciprocity law
  The Shimura reciprocity law is a fundamental result in number theory that generalizes classical reciprocity laws by describing how values of modular functions at complex multiplication (CM) points transform under the action of Galois groups.
• 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. Abelian Varieties with Complex Multiplication and Modular Functions
  "Abelian Varieties with Complex Multiplication and Modular Functions" is a foundational monograph by Goro Shimura that develops the arithmetic theory of abelian varieties with complex multiplication and their deep connections to modular and automorphic functions.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Eichler–Shimura theory
Target entity description: 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.

• A. Shimura correspondence
  The Shimura correspondence is a fundamental result in number theory that establishes a deep link between modular forms of half-integral weight and modular forms of integral weight, with important applications to L-functions and arithmetic geometry.
• B. Shimura varieties
  Shimura varieties are higher-dimensional algebraic varieties that generalize modular curves and play a central role in the Langlands program by connecting number theory, automorphic forms, and arithmetic geometry.
• C. Shimura reciprocity law
  The Shimura reciprocity law is a fundamental result in number theory that generalizes classical reciprocity laws by describing how values of modular functions at complex multiplication (CM) points transform under the action of Galois groups.
• 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. Abelian Varieties with Complex Multiplication and Modular Functions
  "Abelian Varieties with Complex Multiplication and Modular Functions" is a foundational monograph by Goro Shimura that develops the arithmetic theory of abelian varieties with complex multiplication and their deep connections to modular and automorphic functions.
• F. None of above. chosen