Artin's conjecture

Artin's conjecture is a central open prediction in algebraic number theory proposing that nontrivial Artin L-functions attached to irreducible, nontrivial finite-dimensional complex representations of Galois groups are entire. The statement links analytic properties of L-functions with algebraic structures arising from Galois extensions, influencing research across representation theory, algebraic geometry, and arithmetic geometry. The conjecture has driven work connecting the ideas of class field theory, modularity, and automorphic representations.

Statement of the conjecture

The conjecture asserts that for a finite Galois extension E of a number field F with Galois group G and for any irreducible complex representation ρ: G → GL_n(C) that is nontrivial, the associated Artin L-function L(s,ρ) extends to an entire function on the complex plane and satisfies a functional equation. This formulation sits alongside the global reciprocity laws of Emil Artin and complements results from Hecke on abelian L-functions, while contrasting with known poles for the trivial representation linked to the Dedekind zeta function of Dedekind. The conjecture explicitly excludes the trivial representation because of the simple pole at s=1 related to class number and regulator phenomena in Dirichlet's theorem contexts and in the analytic class number formula associated with Gauss and Hilbert.

Known cases and partial results

Proved instances include the one-dimensional (abelian) case resolved by Hecke and refined in the framework of Class field theory developed by Artin and Takagi. The two-dimensional case for odd two-dimensional representations with solvable image was established via the work of Langlands, Tunnell, and Wiles through modularity theorems connecting two-dimensional Galois representations to Modular form L-functions; results invoked the Shimura–Taniyama–Weil conjecture proven for semistable elliptic curves by Wiles and Taylor. Further progress comes from the proof of automorphy lifting theorems by Taylor, Clozel, Harris, Kisin, and Geraghty, which produced analytic continuation for many geometric representations arising from Étale cohomology of varieties studied by Deligne and Grothendieck. The Brauer induction method, introduced by Brauer, reduces Artin L-functions formally to products and quotients of Hecke L-functions, yielding meromorphic continuation for all representations but not holomorphy in general; this approach interacts with ideas from Chebotarev density theorem and Frobenius element distributions first studied by Chebotarev and Frobenius.

Methods and approaches

Approaches to the conjecture divide into analytic, algebraic, and automorphic strategies. The automorphic route relies on the Langlands program and seeks to associate each finite-dimensional Galois representation to an automorphic representation on GL_n over global fields, leveraging the global reciprocity principles envisaged by Langlands and pursued by Jacquet and Shalika. Analytic techniques invoke properties of L-functions developed by Hadamard and Pólya for complex analysis, while algebraic methods exploit Brauer induction, character theory from Burnside, and deformation theory pioneered by Mazur, later expanded by Ramakrishna and Kisin for constructing lifts of mod p representations used in modularity lifting. Geometric methods draw on the cohomological constructions of Grothendieck and the purity results of Deligne in the context of Weil conjectures, translating geometric monodromy into analytic continuation through comparison theorems used by Fontaine and Faltings.

Counterexamples and refinements

No genuine counterexample to the original entireness prediction for nontrivial irreducible complex representations is known. However, the necessity of refinements appears in subtle phenomena: Artin L-functions are known to be meromorphic by Brauer’s theorem, and explicit examples constructed via induced representations show that naive expectations about simple zeros and poles must be adjusted, exemplified in work influenced by Schanuel and investigations into exceptional pole interactions by Davenport and Heilbronn. These examples motivated refined formulations relating possible obstructions to automorphy to the presence of exceptional zeros linked to Siegel zeros in Dirichlet L-series and to phenomena studied by Gross and Stark in Stark's conjectures. The landscape encourages conditional statements: many results assert entireness assuming modularity or potential automorphy hypotheses proven in special contexts by Taylor and collaborators.

Connections to other conjectures and theories

Artin's conjecture sits at the heart of deep conjectural networks: it is a special case of the Langlands reciprocity conjecture connecting Galois representations and automorphic representations, and it complements the Generalized Riemann Hypothesis by specifying analytic continuation and functional equations required for finer zero distribution statements explored by Riemann and Weil. It interacts with the Fontaine–Mazur conjecture predicting which p-adic Galois representations arise from geometry, and with Sato–Tate conjecture type equidistribution results for Frobenius traces proven in cases by Taylor and Harris. Artin's prediction also influences the arithmetic of Elliptic curves, the study of Automorphic forms, and explicit class field constructions related to Kronecker’s Jugendtraum and investigations by Stark and Brumer.

Category:Conjectures in number theory