Atkin–Lehner theory

Target entity: Atkin–Lehner theory
Context triple: [Hecke operators, appearIn, Atkin–Lehner theory]

• A. Hecke theory
  Hecke theory is a branch of number theory centered on Hecke operators and modular forms, providing powerful tools to study arithmetic properties of modular forms and related objects.
• B. 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.
• C. Automorphic Forms and Representations
  Automorphic Forms and Representations is a foundational mathematical monograph that develops the theory of automorphic forms and their connections to representation theory and number theory.
• D. Hecke operators
  Hecke operators are algebraic operators acting on modular forms that play a central role in number theory, particularly in understanding congruences, L-functions, and the arithmetic of modular forms.
• E. 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.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Atkin–Lehner theory
Target entity description: Atkin–Lehner theory is a framework in the theory of modular forms that studies their symmetries and decompositions using certain involutions and operators, refining the structure of spaces of modular forms and newforms.

• A. Hecke theory
  Hecke theory is a branch of number theory centered on Hecke operators and modular forms, providing powerful tools to study arithmetic properties of modular forms and related objects.
• B. 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.
• C. Automorphic Forms and Representations
  Automorphic Forms and Representations is a foundational mathematical monograph that develops the theory of automorphic forms and their connections to representation theory and number theory.
• D. Hecke operators
  Hecke operators are algebraic operators acting on modular forms that play a central role in number theory, particularly in understanding congruences, L-functions, and the arithmetic of modular forms.
• E. 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.
• F. None of above. chosen