Kummer map

Target entity: Kummer map
Context triple: [Kummer theory, relatesTo, Kummer map]

• A. Kummer theory
  Kummer theory is a branch of algebraic number theory that studies abelian extensions of fields, especially cyclotomic and radical extensions, using properties of roots of unity and ideal class groups.
• B. Kummer congruences
  Kummer congruences are number-theoretic relations describing how special values of Bernoulli numbers and related arithmetic functions behave modulo powers of primes, foundational in the study of p-adic L-functions and cyclotomic fields.
• C. Cassels–Tate pairing
  The Cassels–Tate pairing is a bilinear pairing on the Tate–Shafarevich group of an abelian variety over a number field that plays a central role in arithmetic geometry and the study of rational points.
• D. Gross–Koblitz formula
  The Gross–Koblitz formula is a result in number theory that expresses Gauss sums in terms of the p-adic gamma function, linking exponential sums over finite fields with p-adic analysis.
• E. Bloch–Kato conjecture
  The Bloch–Kato conjecture is a deep statement in arithmetic geometry and K-theory that predicts an exact correspondence between Galois cohomology and Milnor K-theory, linking algebraic K-groups to field arithmetic.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Kummer map
Target entity description: The Kummer map is a homomorphism in algebraic number theory and arithmetic geometry that connects elements of a field (or its units) to Galois cohomology classes, playing a key role in describing abelian extensions via Kummer theory.

• A. Kummer theory
  Kummer theory is a branch of algebraic number theory that studies abelian extensions of fields, especially cyclotomic and radical extensions, using properties of roots of unity and ideal class groups.
• B. Kummer congruences
  Kummer congruences are number-theoretic relations describing how special values of Bernoulli numbers and related arithmetic functions behave modulo powers of primes, foundational in the study of p-adic L-functions and cyclotomic fields.
• C. Cassels–Tate pairing
  The Cassels–Tate pairing is a bilinear pairing on the Tate–Shafarevich group of an abelian variety over a number field that plays a central role in arithmetic geometry and the study of rational points.
• D. Gross–Koblitz formula
  The Gross–Koblitz formula is a result in number theory that expresses Gauss sums in terms of the p-adic gamma function, linking exponential sums over finite fields with p-adic analysis.
• E. Bloch–Kato conjecture
  The Bloch–Kato conjecture is a deep statement in arithmetic geometry and K-theory that predicts an exact correspondence between Galois cohomology and Milnor K-theory, linking algebraic K-groups to field arithmetic.
• F. None of above. chosen