Artin L-functions

Artin L-functions
Name	Artin L-functions
Introduced	1923
Founder	Emil Artin
Field	Number theory
Related	L-function, Dedekind zeta function, Hecke L-series

Artin L-functions are complex analytic functions attached to finite-dimensional complex representations of Galois groups of number field extensions, introduced by Emil Artin. They generalize the Dedekind zeta function and connect with objects studied by Ernst Kummer, Bernhard Riemann, Heinrich Weber, and John Tate. Artin L-functions play a central role in modern work of Andrew Wiles, Pierre Deligne, Robert Langlands, Jean-Pierre Serre, and Richard Taylor on reciprocity laws, modularity, and automorphic forms.

Definition and basic properties

For a finite Galois extension K/Q with Galois group G = Gal(K/Q) and a finite-dimensional complex representation ρ: G → GL(V), the Artin L-series is defined initially as an Euler product over primes p of Rudolf Dedekind's decomposition theory. It satisfies multiplicativity for direct sums and compatibility with induction and restriction of representations via the Frobenius reciprocity principle; these features link it to work of Frobenius and Burnside. For the trivial representation one recovers the Dedekind zeta function of K, while for one-dimensional characters one obtains Hecke L-series associated to abelian extensions classified by the Kronecker–Weber theorem and Class field theory developed by David Hilbert and Teiji Takagi.

Construction and local factors

At an unramified prime p the local factor is det(I − ρ(Frob_p) p^{-s})^{-1}, where Frob_p denotes an arithmetic Frobenius element in G determined up to conjugacy; this notion traces to Évariste Galois's permutations and was formalized by Niels Henrik Abel's successors. For ramified primes one uses the action on inertia subgroups and the associated filtration introduced by Herbrand and Shafarevich to define modified local polynomials. The Euler product converges absolutely for Re(s) large because of bounds derived from the finite dimensionality of ρ and eigenvalue estimates related to characters studied by Ferdinand Georg Frobenius and Issai Schur.

Analytic continuation and functional equation

Artin conjectured analytic continuation and functional equation in his foundational papers; the general expectations are a meromorphic continuation to C and a functional equation relating s to 1−s with gamma factors depending on the representation and an epsilon factor (root number). Partial analytic continuation and functional equations are known in cases arising from automorphic representations via the Langlands program and the work of Jacquet, Piatetski-Shapiro, Shalika, Gelbart, Jacquet–Langlands, and Cogdell. Methods exploit the global correspondence between Galois representations and automorphic forms conjectured by Robert Langlands and proven in special cases by Andrew Wiles, Richard Taylor, Michael Harris, Laurent Lafforgue, and Peter Scholze.

Artin conductor and epsilon factors

The Artin conductor measures wild and tame ramification of ρ at each prime, built from ramification filtrations due to Herbrand and Serre; it appears in the exponent of the conductor in the functional equation. The local epsilon factor is a complex root of unity times a power of p^{-s}, influenced by local constants studied by John Tate and Pierre Deligne. Global epsilon factors factor as a product of local epsilon factors and encode subtle reciprocity information related to the Tate–Nakayama duality and the formalism developed by Grothendieck and Serre.

Relation to representation theory and Galois groups

Artin L-functions link linear representations of finite groups with arithmetic invariants of number fields; the use of induced representations refers directly to Frobenius reciprocity and Mackey theory in representation theory of finite groups developed by George Mackey. For extensions with Galois group isomorphic to classical groups studied by Évariste Galois's successors—such as cyclic, dihedral, symmetric, and alternating groups—the structure of irreducible representations (for example those classified by Issai Schur and Frobenius) dictates factorization of L-series. Connections to the Langlands correspondence posit that Artin L-functions should coincide with automorphic L-functions attached to cuspidal representations of reductive groups studied by Harish-Chandra and Jacquet.

Artin conjecture and known results

The Artin conjecture asserts that nontrivial irreducible representations with no trivial subrepresentation produce entire Artin L-functions. Known cases include one-dimensional characters by Hecke and abelian class field theory, two-dimensional odd representations via the modularity theorems of Andrew Wiles, Richard Taylor, and Christopher Skinner building on techniques of Ken Ribet and Frey–Ribet arguments, and solvable-image representations via work of Arthur, Clozel, and Langlands–Tunnell theorem (the latter proved using D. A. Vogan's and Langlands' methods). Many deep results rely on potential automorphy techniques developed by Taylor, Harris, Sheldon Katz, and Kisin.

Examples and special cases

- Abelian extensions: characters correspond to Dirichlet characters and give Dirichlet L-series and Hecke L-series arising from class field theory by Takagi and Artin reciprocity associations. - Quadratic extensions: the two-dimensional permutation representation decomposes yielding factors related to Dirichlet L-series and quadratic characters studied by Gauss and Kronecker. - Tetrahedral, octahedral, and icosahedral Galois groups: representations with images isomorphic to small finite groups studied classically by Galois and Camille Jordan yield L-functions whose analytic behavior was investigated in explicit cases by Langlands, Tunnell, and Buzzard. - Induced representations: permutation representations from subgroups lead to factorization formulas using Frobenius reciprocity and Artin induction.

Category:Number theory