5.3 Artin representations

Artin representations bridge algebraic number theory and representation theory, offering a powerful tool to study Galois groups of number fields. They map abstract group elements to linear transformations, preserving group structure and allowing linear algebra techniques to be applied to abstract group properties.

These representations have finite images, corresponding to Galois groups of finite extensions. Their kernels define normal subgroups, relating to field extensions. The dimension of the representation connects to the degree of the associated field extension, providing insights into field-theoretic properties.

Definition of Artin representations

• Artin representations form a crucial component in Arithmetic Geometry linking algebraic number theory with representation theory
• These representations provide a powerful tool for studying Galois groups of number fields through linear algebra techniques
• Understanding Artin representations lays the foundation for exploring deeper connections in algebraic number theory and representation theory

Group representations basics

Top images from around the web for Group representations basics

• 

  group theory - Visualize Fundamental Homomorphism Theorem for $\phi: A_4 \rightarrow C_3 ... View original

  Is this image relevant?

• 

  Representation Theory [The Physics Travel Guide] View original

  Is this image relevant?

• 

  Linear Algebra – TikZ.net View original

  Is this image relevant?

• 

  group theory - Visualize Fundamental Homomorphism Theorem for $\phi: A_4 \rightarrow C_3 ... View original

  Is this image relevant?

• 

  Representation Theory [The Physics Travel Guide] View original

  Is this image relevant?

1 of 3

Top images from around the web for Group representations basics

• 

  group theory - Visualize Fundamental Homomorphism Theorem for $\phi: A_4 \rightarrow C_3 ... View original

  Is this image relevant?

• 

  Representation Theory [The Physics Travel Guide] View original

  Is this image relevant?

• 

  Linear Algebra – TikZ.net View original

  Is this image relevant?

• 

  group theory - Visualize Fundamental Homomorphism Theorem for $\phi: A_4 \rightarrow C_3 ... View original

  Is this image relevant?

• 

  Representation Theory [The Physics Travel Guide] View original

  Is this image relevant?

1 of 3

• Group representations map abstract group elements to linear transformations on vector spaces
• Consist of a group G and a vector space V over a field K with a homomorphism ρ: G → GL(V)
• Preserve group structure ensuring ρ(gh) = ρ(g)ρ(h) for all g, h ∈ G
• Capture essential information about the group's structure in a linear algebraic setting
• Allow application of linear algebra techniques to study abstract group properties

Galois groups and extensions

• Galois groups emerge as automorphism groups of field extensions
• Describe symmetries of algebraic equations and their roots
• Field extensions K/F create a Galois group Gal(K/F) of field automorphisms fixing F
• Fundamental theorem of Galois theory establishes bijection between intermediate fields and subgroups
• Galois groups of number fields play a central role in Artin representations

Continuous homomorphisms

• Artin representations require continuity in the profinite topology on Galois groups
• Profinite topology defined by inverse limit of finite quotients of the Galois group
• Continuity ensures representation factors through a finite quotient of the Galois group
• Guarantees finite image property of Artin representations
• Allows representation to be studied through finite Galois groups

Properties of Artin representations

• Artin representations bridge abstract Galois theory with concrete linear algebra
• These properties enable powerful techniques from representation theory to be applied to number theoretic problems
• Understanding these properties forms the basis for many important theorems in algebraic number theory

Finite image

• Artin representations always have a finite image due to continuity requirement
• Image is isomorphic to the Galois group of a finite Galois extension
• Allows reduction of infinite Galois groups to finite quotients
• Enables application of finite group theory techniques
• Connects representation theory of infinite profinite groups to finite groups

Kernel and fixed field

• Kernel of an Artin representation defines a normal subgroup of the Galois group
• Fixed field of the kernel corresponds to a finite Galois extension
• Galois correspondence relates subgroups of Gal(K/F) to intermediate fields between K and F
• Kernel determines the field extension associated with the representation
• Studying the kernel provides information about the associated number field extension

Dimension and degree

• Dimension of the representation relates to the degree of the associated field extension
• For irreducible representations, dimension divides the degree of the extension
• Sum of squares of dimensions of irreducible components equals the extension degree
• Provides a way to decompose field extensions into simpler components
• Connects linear algebraic properties (dimension) with field-theoretic properties (degree)

Classification of Artin representations

• Classification of Artin representations provides a systematic way to understand Galois groups
• This classification forms the basis for many important results in algebraic number theory
• Understanding the structure of Artin representations is crucial for applications in the Langlands program

One-dimensional representations

• Simplest type of Artin representations corresponding to abelian extensions
• Characterized by their kernel which determines a cyclic extension of the base field
• Closely related to Dirichlet characters and ideal class groups
• Play a crucial role in class field theory and the proof of quadratic reciprocity
• Can be composed to create higher-dimensional abelian representations

Irreducible representations

• Fundamental building blocks of all Artin representations
• Cannot be decomposed into simpler representations
• Correspond to minimal normal subgroups of the Galois group
• Dimensions of irreducible representations divide the order of the Galois group
• Schur's lemma provides powerful tools for studying irreducible representations

Decomposition into irreducibles

• Every Artin representation can be uniquely decomposed into irreducible components
• Decomposition reflects the structure of the associated Galois extension
• Multiplicities in the decomposition provide important arithmetic information
• Artin's induction theorem relates any representation to sum of induced one-dimensional representations
• Decomposition process often reveals hidden symmetries in the Galois group

Artin L-functions

• Artin L-functions generalize Dirichlet L-functions to non-abelian extensions
• These functions encode deep arithmetic information about number fields and their Galois groups
• Understanding Artin L-functions is crucial for many open problems in number theory

Definition and properties

• Artin L-functions defined as Euler products over prime ideals of the base field
• Local factors at each prime determined by the representation's behavior on Frobenius elements
• Incorporate information about ramification and splitting of primes in the extension
• Satisfy a functional equation relating values at s and 1-s
• Conjectured to have meromorphic continuation to the entire complex plane

Analytic continuation

• Artin L-functions initially defined only for Re(s) > 1
• Analytic continuation to the whole complex plane is a major open problem (Artin's conjecture)
• Proved for one-dimensional representations (abelian case) using class field theory
• Known for some specific non-abelian cases (solvable groups)
• Langlands program aims to prove continuation for all Artin L-functions

Functional equation

• Artin L-functions satisfy a functional equation relating L(s,ρ) and L(1-s,ρ*)
• Involves gamma factors depending on the signature of the representation
• Reflects deep symmetries in the arithmetic of number fields
• Crucial for understanding the behavior of L-functions on the critical line Re(s) = 1/2
• Analogous to functional equations for zeta functions and modular forms

Artin's reciprocity law

• Artin's reciprocity law generalizes quadratic reciprocity to arbitrary abelian extensions
• This law forms the cornerstone of class field theory, connecting Galois theory with algebraic number theory
• Understanding Artin reciprocity is crucial for many advanced topics in arithmetic geometry

Statement of the law

• Establishes isomorphism between norm quotient group and Galois group for abelian extensions
• Generalizes quadratic reciprocity law to arbitrary abelian extensions
• Provides explicit description of the Artin map from idele class group to Galois group
• Relates local and global class field theory through compatibility with local reciprocity laws
• Forms the basis for the Langlands reciprocity conjecture in the general case

Abelian case vs general case

• Abelian case fully resolved by class field theory
• General non-abelian case remains an open problem (Langlands reciprocity conjecture)
• Abelian case involves commutative Galois groups and one-dimensional representations
• Non-abelian case requires understanding of higher-dimensional representations
• Transition from abelian to non-abelian case involves deep connections with automorphic forms

Connection to class field theory

• Artin reciprocity provides the main theorem of global class field theory
• Establishes bijection between abelian extensions and norm subgroups of the idele class group
• Allows explicit construction of class fields using ray class groups
• Provides a way to study ramification in abelian extensions
• Forms the basis for generalizations to non-abelian class field theory

Applications in number theory

• Artin representations and their associated L-functions have numerous applications in number theory
• These applications range from concrete results about prime numbers to deep conjectures in the Langlands program
• Understanding these applications provides insight into the power and versatility of Artin representations

Chebotarev density theorem

• Generalizes Dirichlet's theorem on primes in arithmetic progressions
• States that Frobenius elements are equidistributed in the Galois group
• Provides information about splitting of primes in Galois extensions
• Has applications in inverse Galois theory and the distribution of prime ideals
• Proof relies heavily on properties of Artin L-functions

Solvability of Galois groups

• Artin representations provide tools for studying solvability of Galois groups
• Relate to Galois cohomology and embedding problems
• Allow investigation of inverse Galois problem for certain types of groups
• Connect to modular forms and elliptic curves in special cases
• Provide insights into the structure of absolute Galois groups of number fields

Langlands program connections

• Artin representations form a key part of the Langlands correspondence
• Conjectured to correspond to automorphic representations of GL(n)
• Relate to modular forms and more general automorphic forms
• Play a role in the proof of Serre's modularity conjecture
• Provide a bridge between number theory and representation theory of adelic groups

Computational aspects

• Computational techniques for Artin representations are crucial for concrete applications
• These methods allow for explicit calculations and verifications of theoretical results
• Understanding computational aspects provides practical tools for working with Artin representations

Algorithms for Artin representations

• Involve computing Galois groups of polynomials over number fields
• Require efficient methods for handling large finite groups
• Include techniques for decomposing representations into irreducible components
• Utilize character theory and modular arithmetic for calculations
• Employ linear algebra algorithms adapted to representation theory

Software tools and packages

• PARI/GP provides functions for working with Artin representations
• SageMath includes modules for Galois theory and representation theory
• GAP offers extensive capabilities for group theory and representation theory
• Magma has specialized functions for Artin representations and L-functions
• LMFDB (L-functions and Modular Forms Database) contains tables of Artin representations

Complexity considerations

• Computing Galois groups has exponential worst-case complexity
• Decomposing representations into irreducibles can be done in polynomial time
• Evaluating Artin L-functions requires efficient algorithms for handling Euler products
• Verifying Artin's conjecture computationally is extremely challenging for large degrees
• Trade-offs between time complexity and space complexity in storing precomputed data

Examples and special cases

• Studying specific examples and special cases of Artin representations provides concrete understanding
• These cases often illustrate general principles and highlight important features
• Examining examples is crucial for developing intuition about Artin representations

Quadratic extensions

• Simplest non-trivial case of Artin representations
• Correspond to quadratic characters and Legendre symbols
• Related to quadratic reciprocity law and genus theory
• L-functions in this case are Dirichlet L-functions for quadratic characters
• Provide a bridge between elementary and advanced aspects of algebraic number theory

Cyclic extensions

• Generalize quadratic case to arbitrary prime power degrees
• Representations decompose into one-dimensional characters
• Closely related to cyclotomic fields and Kummer theory
• L-functions factor as products of Dirichlet L-functions
• Illustrate principles of abelian class field theory

Non-abelian examples

• Include representations of symmetric and alternating groups
• Arise from Galois groups of polynomials (cubic, quartic, quintic equations)
• Provide concrete cases where Artin's conjecture is known to hold
• Relate to modular forms in low-dimensional cases (Langlands correspondence)
• Illustrate challenges in generalizing abelian theory to non-abelian case

Generalizations and variations

• Artin representations have inspired numerous generalizations and variations
• These extensions connect Artin's ideas to broader areas of mathematics
• Understanding these generalizations provides insight into the far-reaching impact of Artin representations

l-adic representations

• Generalize Artin representations to l-adic coefficients
• Arise naturally in the study of étale cohomology of varieties
• Play crucial role in modern algebraic geometry and arithmetic geometry
• Connect to Galois representations associated to elliptic curves and modular forms
• Form basis for many results in arithmetic of abelian varieties

Motivic Galois representations

• Extend idea of Galois representations to motives
• Conjecturally provide unified framework for all "natural" Galois representations
• Relate to Hodge structures and periods of algebraic varieties
• Connect to Grothendieck's theory of motives and motivic cohomology
• Play central role in formulation of many deep conjectures in arithmetic geometry

Geometric Langlands program

• Translates ideas of Artin representations and Langlands program to algebraic geometry
• Replaces number fields with function fields of curves over finite fields
• Involves representations of algebraic groups and D-modules on moduli stacks
• Connects representation theory, algebraic geometry, and mathematical physics
• Provides geometric interpretation of automorphic forms and Hecke operators