Mathematics Branches, Topics, and Sub-Topics

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19Fxx K-theory in number theory and arithmetic

This subtopic studies K-theory in number theory and arithmetic, linking algebraic K-groups with arithmetic invariants and special values of L-functions.

Specific topics

19F05 Generalized class field theory

Overview

19F05 studies generalized class field theory in k-theory in number theory and arithmetic. It connects K-theoretic invariants with arithmetic objects such as number fields, zeta values, regulators, and étale cohomology.

Related Wikipedia Page

K-Theory In Number Theory And Arithmetic (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for generalized class field theory
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to study arithmetic invariants, regulators, and special values of L-functions.

Applications

  • Number-theoretic K-theory
  • Regulators and special values
  • Étale and arithmetic applications

References

Recommended Textbooks

19F10 Étale cohomology, higher regulators, zeta and $L$-functions

Overview

19F10 studies etale cohomology, higher regulators, zeta and $l$-functions in k-theory in number theory and arithmetic. It connects K-theoretic invariants with arithmetic objects such as number fields, zeta values, regulators, and étale cohomology.

Related Wikipedia Page

K-Theory In Number Theory And Arithmetic (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for etale cohomology, higher regulators, zeta and $l$-functions
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to study arithmetic invariants, regulators, and special values of L-functions.

Applications

  • Number-theoretic K-theory
  • Regulators and special values
  • Étale and arithmetic applications

References

Recommended Textbooks

19F15 Symbols and arithmetic

Overview

19F15 studies symbols and arithmetic in k-theory in number theory and arithmetic. It connects K-theoretic invariants with arithmetic objects such as number fields, zeta values, regulators, and étale cohomology.

Related Wikipedia Page

K-Theory In Number Theory And Arithmetic (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for symbols and arithmetic
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to study arithmetic invariants, regulators, and special values of L-functions.

Applications

  • Number-theoretic K-theory
  • Regulators and special values
  • Étale and arithmetic applications

References

Recommended Textbooks

19F27 Étale cohomology and $K$-theory of fields

Overview

19F27 studies etale cohomology and $k$-theory of fields in k-theory in number theory and arithmetic. It connects K-theoretic invariants with arithmetic objects such as number fields, zeta values, regulators, and étale cohomology.

Related Wikipedia Page

K-Theory In Number Theory And Arithmetic (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for etale cohomology and $k$-theory of fields
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to study arithmetic invariants, regulators, and special values of L-functions.

Applications

  • Number-theoretic K-theory
  • Regulators and special values
  • Étale and arithmetic applications

References

Recommended Textbooks