12.3 Artin L-functions and reciprocity laws

Artin L-functions are powerful tools in number theory, encoding arithmetic information about Galois representations. They're defined as infinite products over prime ideals and reveal deep connections between algebra and analysis through their properties.

The Artin reciprocity law bridges number theory and representation theory, equating certain L-functions. Its proof uses algebraic techniques and has far-reaching implications, generalizing reciprocity laws and laying groundwork for modern number theory advances.

Artin L-functions and Galois Representations

Artin L-functions of Galois representations

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• Artin L-functions define infinite product over prime ideals encoding arithmetic information about Galois representations
  • $L(s, \rho, K/F) = \prod_{p} \det(I - \rho(\text{Frob}_p) N(p)^{-s})^{-1}$
  • $\rho$ represents Galois group $\text{Gal}(K/F)$ (group of field automorphisms)
  • $K/F$ denotes Galois extension of number fields (algebraic number fields with Galois group)
  • $p$ runs over prime ideals of $F$ (ideals generated by irreducible elements)
  • $\text{Frob}_p$ signifies Frobenius element at $p$ (special automorphism in Galois group)
  • $N(p)$ calculates norm of prime ideal $p$ (number of elements in residue field)
• Properties of Artin L-functions reveal deep connections between algebra and analysis
  • Meromorphic continuation extends function beyond initial domain of convergence
  • Functional equation relates values of L-function at $s$ and $1-s$
  • Analytic behavior at $s=1$ connects to important arithmetic properties (class numbers, regulators)
• Relation to Dedekind zeta functions demonstrates factorization principle
  • $\zeta_K(s) = \prod_{\rho} L(s, \rho, K/F)^{\dim \rho}$
  • Decomposes zeta function of larger field into L-functions of smaller field

Proof of Artin reciprocity law

• Artin reciprocity law establishes fundamental connection between number theory and representation theory
  • $L(s, \chi, K/F) = L(s, \rho_\chi, K/F)$
  • $\chi$ represents Hecke character of $F$ (generalization of Dirichlet characters)
  • $\rho_\chi$ denotes representation of $\text{Gal}(K/F)$ induced by $\chi$
• Representation-theoretic proof utilizes powerful algebraic techniques
  1. Apply Brauer's theorem on induced characters to reduce to cyclic case
  2. Use properties of cyclic extensions to further simplify
  3. Leverage class field theory for cyclic extensions to complete proof
• Implications of Artin reciprocity extend far beyond original context
  • Generalizes quadratic reciprocity to arbitrary abelian extensions
  • Bridges Hecke L-functions and Artin L-functions
  • Provides foundation for ambitious Langlands program in modern number theory

Applications to class field theory

• Class field theory describes structure of abelian extensions of number fields
  • Existence theorem establishes correspondence between abelian extensions and norm subgroups
  • Uniqueness theorem ensures one-to-one nature of this correspondence
• Structure of abelian extensions revealed through powerful formulas
  • Conductor-discriminant formula relates important invariants of extensions
  • Kronecker-Weber theorem characterizes abelian extensions of rational numbers (cyclotomic fields)
• Applications of class field theory solve classical problems in algebraic number theory
  • Hilbert class field corresponds to ideal class group (measures how far ring of integers is from unique factorization)
  • Ray class fields generalize ideal class groups to more refined invariants
  • Galois groups of abelian extensions computed using class field theory machinery

Connections with zeta functions

• Zeta functions encode deep arithmetic information about number fields
  • Riemann zeta function $\zeta(s)$ fundamental to study of prime numbers
  • Dedekind zeta function $\zeta_K(s)$ generalizes to arbitrary number fields
• Distribution of prime ideals closely tied to properties of L-functions
  • Chebotarev density theorem describes equidistribution of prime ideals
  • Frobenius elements in Galois group correspond to splitting behavior of primes
• Connections between L-functions and zeta functions reveal underlying structure
  • Dedekind zeta functions factor into product of Artin L-functions
  • Analytic class number formula relates special values of zeta functions to important field invariants
  • Brauer-Siegel theorem bounds growth of class numbers in terms of discriminants
• Applications to number theory solve classical problems
  • Dirichlet's theorem on primes in arithmetic progressions follows from properties of L-functions
  • Density of splitting primes in extensions computed using Chebotarev density theorem
  • Effective versions of Chebotarev density theorem provide explicit bounds and error terms