Eisenstein ideal

Target entity: Eisenstein ideal
Context triple: [Barry Mazur, knownFor, Eisenstein ideal]

• A. Dedekind ideal
  A Dedekind ideal is a type of ideal in ring theory central to algebraic number theory, particularly in the study of Dedekind domains and unique factorization of ideals.
• B. Artin reciprocity law
  The Artin reciprocity law is a fundamental theorem in class field theory that generalizes quadratic reciprocity by describing abelian extensions of number fields in terms of characters of their idele class groups.
• 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. Iwasawa theory
  Iwasawa theory is a branch of number theory that studies the growth of arithmetic invariants in infinite towers of number fields, particularly using p-adic methods.
• E. 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.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Eisenstein ideal
Target entity description: The Eisenstein ideal is a specific ideal in the Hecke algebra of modular forms that plays a central role in the study of congruences between modular forms and the arithmetic of elliptic curves.

• A. Dedekind ideal
  A Dedekind ideal is a type of ideal in ring theory central to algebraic number theory, particularly in the study of Dedekind domains and unique factorization of ideals.
• B. Artin reciprocity law
  The Artin reciprocity law is a fundamental theorem in class field theory that generalizes quadratic reciprocity by describing abelian extensions of number fields in terms of characters of their idele class groups.
• 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. Iwasawa theory
  Iwasawa theory is a branch of number theory that studies the growth of arithmetic invariants in infinite towers of number fields, particularly using p-adic methods.
• E. 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.
• F. None of above. chosen