Artin representation

Artin representation
Name	Artin representation
Field	Algebraic number theory, Representation theory, Galois theory
Introduced	1930s
Introduced by	Emil Artin
Related	Galois representation, Artin L-function, Langlands program, Class field theory

Artin representation is a finite-dimensional complex linear representation of a Galois group of a finite extension of number fields, encoding arithmetic and arithmetic‑geometric information via group actions on complex vector spaces. These representations, introduced by Emil Artin, link key objects in algebraic number theory, representation theory, and the emerging framework of the Langlands program. They play central roles in the study of Artin L-functions, ramification in extensions studied by Frobenius elements, and reciprocity laws from Class field theory.

Definition and basic properties

An Artin representation is a continuous homomorphism from the absolute or finite extension Galois group G = Gal(E/F) of a finite extension E/F of number fields into GL_n(C), where continuity refers to the profinite topology on G and the discrete topology on GL_n(C). Typical finite sources include Gal(E/F) for E a finite Galois extension of Q or other number fields such as Q(√2), Q(ζ_n), or extensions arising from elliptic curve torsion. Basic invariants include the representation degree n, character χ: G → C, and the associated field of definition generated by character values, intimately related to cyclotomic fields like Q(ζ_m). Characters satisfy orthogonality relations familiar from Frobenius–Schur indicator and decompose into irreducible constituents corresponding to irreducible representations of finite groups such as S_n, A_n, D_n, GL_2(F_p) and Heisenberg group quotients.

Local and global Artin representations

Artin representations admit local and global incarnations: globally as representations of Gal(E/F) for a global base F like Q, Q(√-1), Q(√5), or totally real field extensions; locally as representations of decomposition groups Gal(E_v/F_v) at places v corresponding to finite primes like p-adic completions Q_p or real/complex places such as R and C. The passage between global and local data uses restriction maps to inertia and decomposition subgroups attached to primes like Prime ideals over p in rings of integers of E. Local representations are classified by ramification behavior detected by the filtration of inertia groups, connecting to local objects such as Weil groups and Weil–Deligne groups when relating to automorphic representations on local groups like GL_n(Q_p).

Artin L-functions

To each finite-dimensional Artin representation ρ: Gal(E/F) → GL_n(C) one associates an analytic function L(s,ρ) defined by an Euler product over places v of F. At an unramified finite place v with Frobenius conjugacy class Fr_v ∈ Gal(E/F), the local Euler factor is det(1 − ρ(Fr_v) N(v)^{-s})^{-1}, where N(v) denotes the norm of the prime as in Absolute norm. The collection of these factors yields the global Artin L-function L(s,ρ), which generalizes Dedekind zeta functions and intertwines with known L-functions like those of Dirichlet characters, Hecke characters, and L-series of elliptic curves. Conjectures in the Langlands program—including Artin’s conjecture on holomorphy—predict that nontrivial irreducible ρ with no trivial subrepresentation have entire L(s,ρ) and correspond to automorphic L-functions for groups such as GL_n(A_F), where A_F denotes the adele ring of F.

Conductors and ramification

The conductor of an Artin representation measures its ramification across primes, combining exponents from wild and tame components via the Artin conductor formula. For a local representation at a prime p the conductor exponent uses the filtration by higher ramification groups (upper numbering) originally studied by Herbrand and formalized by Serre; globally, the conductor is an ideal of the base field built from local exponents. Ramification behavior is reflected in local epsilon factors entering the functional equation of L(s,ρ), linking to objects like the Tate local epsilon factor and Local Langlands correspondence predictions. Wild ramification arises in extensions with nontrivial p-Sylow inertia such as certain cyclotomic extensions and nonabelian extensions with Galois groups like S_3 or D_p.

Examples and classification

Basic examples include 1-dimensional Artin representations given by characters Gal(E/F) → C× arising from abelian extensions classified by Class field theory; these yield Dirichlet L-functions and Hecke L-functions. Two-dimensional examples appear from permutation representations of groups like S_3 leading to tetrahedral cases, or from induced representations from index-two subgroups producing dihedral types linked with D_n groups. Irreducible 2-dimensional complex representations of groups like A_5 produce icosahedral Artin representations tied to nonabelian extensions such as those constructed via Quintic equations studied by Klein. Higher-dimensional cases occur by induction from subgroups or via tensor operations combining representations associated to groups like GL_2(F_p) and finite simple groups including PSL_2(F_q). Classification leverages the Brauer induction theorem and character theory for finite groups like C_n, S_n, A_n, Q_8, SL_2(F_p).

Relations to Galois and automorphic representations

Artin representations form a bridge between finite Galois theory and analytic objects in the Langlands program, conjecturally matching irreducible complex representations of Gal(E/F) with automorphic representations of GL_n(A_F) or other reductive groups. Proven instances include the correspondence for 1-dimensional characters via Class field theory and the modularity of certain 2-dimensional odd representations via theorems of Andrew Wiles, Richard Taylor, and Freitas–Le Hung–Siksek for elliptic curves and modular forms like those arising from Taniyama–Shimura–Weil conjecture. The study of Artin representations interfaces with work on potential automorphy by Clozel, Harris, and Taylor, and with reciprocity laws exemplified by Langlands reciprocity conjectures connecting Artin L-functions to automorphic L-functions and thus to objects such as modular forms, Maass forms, and automorphic representations for classical groups.

Category:Algebraic number theory