15.3 Applications of class field theory

Class field theory is a powerful tool in algebraic number theory. It describes abelian extensions of global and local fields, providing insights into field arithmetic and solving complex problems in number theory and arithmetic geometry.

This theory enables us to construct and analyze important objects like the Hilbert class field. It also has practical applications in solving Diophantine equations, computing class numbers, and studying prime ideal distribution in number fields.

Class Field Theory Applications

Solving Problems in Algebraic Number Theory and Arithmetic Geometry

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• Class field theory comprehensively describes abelian extensions of global and local fields
  • Serves as a powerful tool for solving various problems in algebraic number theory and arithmetic geometry
• Artin reciprocity law establishes correspondence between abelian extensions of a number field and quotients of its idele class group
• Enables study of the Galois group of the maximal abelian extension of a number field
  • Provides insights into the field's arithmetic properties
• Hilbert class field constructed and analyzed using class field theory techniques
  • Maximal unramified abelian extension of a number field
• Applications in arithmetic geometry include:
  • Study of rational points on elliptic curves
  • Investigation of modular forms
• Provides tools for understanding behavior of prime ideals in abelian extensions
  • Splitting properties
  • Ramification properties
• Theory of complex multiplication combines class field theory with elliptic curve theory
  • Significant applications in constructing abelian extensions of imaginary quadratic fields

Practical Applications and Problem-Solving

• Solves Diophantine equations using abelian extensions
• Determines solvability of certain algebraic equations over number fields
• Computes class numbers and unit groups of number fields
• Constructs number fields with specific Galois groups
• Analyzes the distribution of prime ideals in number fields
• Proves reciprocity laws (quadratic, cubic, biquadratic)
• Studies the arithmetic of elliptic curves over number fields

Structure of Abelian Extensions

Fundamental Theorems and Concepts

• Existence Theorem guarantees unique abelian extension for every open subgroup of finite index in the idele class group
• Artin map establishes isomorphism between:
  • Galois group of an abelian extension
  • Quotient of the idele class group of the base field
• Provides complete classification of all finite abelian extensions of a given number field
  • Achieved through study of its idele class group and quotients
• Conductor of an abelian extension precisely described using class field theory
  • Measures ramification of the extension
• Local class field theory focuses on local fields
  • Provides foundation for understanding local behavior of abelian extensions
  • Relates local extensions to global extensions

Detailed Analysis of Abelian Extensions

• Decomposition groups of prime ideals in abelian extensions explicitly described
  • Provides insights into arithmetic of these extensions
• Inertia groups of prime ideals in abelian extensions characterized
• Explicit construction of certain types of abelian extensions
  • Ray class fields generalize the concept of the Hilbert class field
• Norm residue symbol used to describe abelian extensions
• Frobenius elements in abelian extensions studied using class field theory
• Reciprocity laws derived from class field theory
• Genus theory of quadratic fields explained through class field theory

Zeta and L-functions with Class Field Theory

Factorization and Analytic Properties

• Class field theory provides framework for understanding factorization of Dedekind zeta function
  • Dedekind zeta function of a number field factors into product of Artin L-functions
• Artin reciprocity law allows interpretation of Artin L-functions as automorphic L-functions
  • Leads to important analytic properties (analytic continuation, functional equation)
• Enables study of L-function behavior at special values
  • Relates to important arithmetic invariants of number fields (class numbers, regulators)
• Theory of complex multiplication provides tools for studying L-functions of elliptic curves with complex multiplication
• Allows investigation of Riemann hypothesis for function fields
  • Provides insights potentially applicable to classical Riemann hypothesis

Advanced Topics and Special Values

• Study of p-adic L-functions deeply connected to class field theory and Iwasawa theory
  • p-adic L-functions are p-adic analogues of classical L-functions
• Provides methods for studying special values of L-functions
  • Relates to periods of motives and other arithmetic objects
• Stark conjectures on special values of L-functions explored using class field theory
• Birch and Swinnerton-Dyer conjecture studied through class field theory techniques
• Gross-Zagier formula relating L-functions to heights of Heegner points explained using class field theory
• Kummer's criterion for divisibility of class numbers by primes derived from class field theory
• Iwasawa main conjecture relates p-adic L-functions to ideal class groups

Class Field Theory in Cryptography vs Coding Theory

Cryptographic Applications

• Provides theoretical foundation for public-key cryptosystems based on discrete logarithms in finite fields
• Theory of complex multiplication used in construction of elliptic curves for elliptic curve cryptography
  • Allows creation of curves with specific properties
• Tools for understanding and constructing certain types of cryptographic hash functions
• Study of division fields of elliptic curves applies to pairing-based cryptography
• Contributes to understanding of lattice-based cryptosystems
  • Potential candidates for post-quantum cryptography
• Helps in designing protocols for secure multi-party computation
• Provides basis for homomorphic encryption schemes

Coding Theory Applications

• Class field towers used in construction of asymptotically good families of codes
  • Infinite sequences of unramified extensions studied through class field theory
• Explicit class field theory of global function fields applied to algebraic-geometric codes
  • Constructs codes with good parameters
• Goppa codes, a class of linear error-correcting codes, understood through class field theory
• Class field theory techniques used in decoding algorithms for certain algebraic codes
• Provides tools for analyzing the weight distribution of certain codes
• Helps in constructing codes with specific automorphism groups
• Applied in the study of quantum error-correcting codes based on algebraic structures