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Artin L -function

In mathematics, particularly in algebraic number theory, an Artin L-function is a Dirichlet series defined for a finite-dimensional complex representation \rho of the Galois group \Gal(L/K) of a finite Galois extension L/K of number fields, generalizing the classical Dirichlet and Dedekind zeta functions to non-abelian settings.^[1] It takes the form L(s, \rho) = \prod_p \det(I - N(p)^{-s} \rho(\Frob_p))^{-1}, where the product runs over unramified primes p of K, N(p) is the norm of p, and \Frob_p is the Frobenius element, with local factors extended to ramified primes via decomposition and inertia groups.^[1] Introduced by Emil Artin in 1923 during his work on non-abelian class field theory, these functions encode arithmetic information about the extension L/K through the representation \rho, and they satisfy key properties such as multiplicativity and a functional equation relating L(s, \rho) to L(1-s, \rho^\vee), where \rho^\vee is the contragredient representation.^[1]^[2] Artin's motivation stemmed from decomposing the Dedekind zeta function \zeta_L(s) of the extension as \zeta_L(s) = \zeta_K(s) \prod_\rho L(s, \rho)^{m_\rho}, where the product is over irreducible representations of \Gal(L/K) with multiplicities m_\rho, aiming to explain the poles and analytic behavior of zeta functions in terms of these L-functions.^[1] Building on earlier work by Dirichlet (for characters of \mathbb{Z}^\times) and Hecke (for Grössencharacters), Artin extended the theory to arbitrary finite-dimensional representations, proving in 1923 that the L-functions converge absolutely for \Re(s) > 1 and are multiplicative under tensor products of representations.^[1] For abelian extensions, where \rho is one-dimensional, Artin's L-functions coincide with Hecke L-functions, whose analytic continuation and functional equations were established via class field theory.^[2] The analytic properties of Artin L-functions remain a central focus of research, with Artin's 1930 conjecture positing that for irreducible, non-trivial representations \rho, L(s, \rho) extends to an entire function on the complex plane (except possibly for the trivial representation, which introduces the pole of the zeta function).^[1] This conjecture implies the holomorphy of \zeta_L(s) away from its known pole at s=1 and has been proven in special cases, such as for dihedral representations by Langlands in the 1970s and more generally under GRH by partial results, but remains open in full generality.^[1] Through the Langlands program, Artin L-functions connect to automorphic forms, with the Artin conjecture equivalent to the modularity of the corresponding Galois representations, highlighting their role in unifying algebraic and analytic number theory.^[1]

Background and Definition

Historical Context

The Artin L-function was introduced by Emil Artin in 1923 as part of his efforts to develop a non-abelian class field theory, generalizing the existing framework for abelian extensions to arbitrary finite Galois extensions of number fields.^[3] In his seminal paper "Über eine neue Art von L-Reihen," Artin defined these functions in terms of characters of the Galois group, motivated by the need to extend Dirichlet L-functions—which encode arithmetic data for abelian characters over the rationals—to Galois representations over general number fields, thereby capturing splitting behavior of primes in non-abelian settings.^[4] This innovation aimed to bridge algebraic structures like Galois groups with analytic tools, providing a means to study reciprocity laws beyond the abelian case.^[1] Artin's work built upon foundational contributions from earlier mathematicians in class field theory and representation theory. Ferdinand Georg Frobenius's studies on the density of primes and induced representations of finite groups, particularly through Frobenius substitutions and reciprocity for characters, directly informed Artin's construction of Euler factors for L-functions using Galois actions on primes.^[1] Richard Dedekind's introduction of zeta functions for number fields and his early insights into ideal class groups provided the arithmetic backbone, while Teiji Takagi's rigorous completion of abelian class field theory in the early 1920s—establishing the isomorphism between ideal class groups and Galois groups for abelian extensions—inspired Artin to pursue analogous results for non-abelian cases.^[5] These influences shaped Artin's vision of L-functions as a unifying analytic device for class field theory. Key advancements occurred in Artin's papers during the 1930s, where he refined the theory by incorporating induced representations and linking it to broader reciprocity principles. In 1930, Artin addressed incomplete aspects of his initial definition, such as handling ramified primes and infinite places, by introducing the Artin conductor and proving properties like inductivity, which equates the L-function of an induced representation to that of the original one via Frobenius reciprocity.^[1] His 1931 paper further connected these L-functions to the general reciprocity law he had established in 1927, demonstrating how they encode the decomposition of Dedekind zeta functions into Artin factors and facilitating proofs of density theorems for splitting primes.^[3] These developments solidified the role of Artin L-functions in non-abelian extensions, setting the foundation for subsequent analytic investigations.^[6]

Formal Definition

The Artin L-function is defined in the context of Galois representations over number fields. Let K be a number field, and let G = \mathrm{Gal}(\overline{\mathbb{Q}}/K) denote its absolute Galois group. An Artin representation \rho: G \to \mathrm{GL}_n(\mathbb{C}) is a continuous, finite-dimensional representation, typically assumed to be irreducible for the primitive case.^[7] This setup generalizes the earlier constructions of Dirichlet and Hecke L-functions to non-abelian Galois extensions.^[4] The L-function attached to such a representation \rho is given by the Euler product formula over the finite primes:

L(s, \rho) = \prod_{\mathfrak{p}} \det\left(I - \rho(\mathrm{Frob}_{\mathfrak{p}}) \, N(\mathfrak{p})^{-s} \,\middle|\, V^\rho \right)^{-1},

where the product runs over all prime ideals \mathfrak{p} of the ring of integers \mathcal{O}_K unramified with respect to \rho (i.e., prime ideals for which the inertia group acts trivially on the representation space V^\rho), \mathrm{Frob}_{\mathfrak{p}} is an element of the Frobenius conjugacy class in G associated to \mathfrak{p}, and N(\mathfrak{p}) is the absolute norm of \mathfrak{p}.^[7] For ramified prime ideals \mathfrak{p}, the corresponding local factors are defined using the subspace of V^\rho fixed by the inertia subgroup I_{\mathfrak{p}}, specifically L_{\mathfrak{p}}(s, \rho) = \det\left(I - \rho(\mathrm{Frob}_{\mathfrak{p}}) \, N(\mathfrak{p})^{-s} \,\middle|\, (V^\rho)^{I_{\mathfrak{p}}} \right)^{-1}, ensuring the product converges absolutely for \mathrm{Re}(s) > 1.^[7] This construction extends Artin's original 1923 definition, which used the trace of the representation (character) to form the Dirichlet series, equivalently yielding the determinant form via properties of matrices.^[4] For reducible representations, the L-function multiplicatively decomposes according to the direct sum structure: if \rho = \bigoplus_i m_i \rho_i where the \rho_i are distinct irreducibles with multiplicities m_i, then L(s, \rho) = \prod_i L(s, \rho_i)^{m_i}.^[7] This follows from the determinant property for block-diagonal matrices. The irreducibility components are uniquely determined by the orthogonality of irreducible characters under the inner product on the group algebra of G, which ensures the distinct L-functions associated to different irreducibles are analytically independent in their factorizations.^[7] A key example arises when \rho is one-dimensional, corresponding to a continuous character \chi: G \to \mathbb{C}^\times. In this case, L(s, \chi) reduces to a Hecke L-function attached to the abelian extension of K cut out by the kernel of \chi, generalizing the classical Dirichlet L-functions over \mathbb{Q}.^[7] For instance, if K = \mathbb{Q} and \chi is the trivial character, then L(s, \chi) = \zeta(s), the Riemann zeta function.^[4]

Analytic Properties

Functional Equation

The completed Artin L-function for an irreducible representation \rho of the Galois group G = \mathrm{Gal}(K/\mathbb{Q}), where K/\mathbb{Q} is a finite Galois extension with discriminant N = |d_K|, is defined as

\Lambda(s, \rho) = N^{s/2} \left( \prod_v \Gamma_v(s, \rho) \right) L(s, \rho),

where the product runs over all places v of \mathbb{Q}, the local gamma factors \Gamma_v(s, \rho) are 1 at finite places, and L(s, \rho) is the Artin L-function.^[8]^[1] At infinite places, the gamma factors \Gamma_\infty(s, \rho) depend on the action of the decomposition group (generated by complex conjugation for real embeddings). For the real infinite place of \mathbb{[Q](/page/Q)}, if c denotes complex conjugation acting on \rho, let

V^{&lt;c&gt;} = \{ v \in V \mid \rho(c)v = v \}

(the +1 eigenspace) and V^c = \{ v \in V \mid \rho(c)v = -v \} (the -1 eigenspace). Then

\Gamma_R(s, \rho) = \Gamma\left(\frac{s}{2}\right)^{\dim V^{&lt;c&gt;}} \Gamma\left(\frac{s+1}{2}\right)^{\dim V^c}.

For precision, the full factors are

[\pi^{-s/2} \Gamma(s/2)]^{\dim V^{&lt;c&gt;}} [\pi^{-(s+1)/2} \Gamma((s+1)/2)]^{\dim V^c}

. If the base field has complex infinite places, the gamma factor for each is (2(2\pi)^{-s} \Gamma(s))^{\dim \rho}, but for base \mathbb{[Q](/page/Q)}, only the real case applies.^[1]^[6] The root number \varepsilon(\rho) is the product of local epsilon factors \varepsilon_v(\rho, \psi_v) over all places v, where \psi_v is a fixed nontrivial additive character of the local field; it satisfies |\varepsilon(\rho)| = 1.^[8]^[1] The functional equation takes the form

\Lambda(s, \rho) = \varepsilon(\rho) \, N^{1/2 - s} \, \Lambda(1 - s, \bar{\rho}),

where \bar{\rho} is the contragredient (complex conjugate) representation; this relates the completed L-function at s to its value at $1 - s.^[8]^[6] The equation implies that \Lambda(s, \rho) admits meromorphic continuation to the entire complex plane \mathbb{C}, holomorphic except possibly at s = 1 if \rho contains the trivial representation.^[8]^[1] The proof proceeds by Brauer's induction theorem, which expresses the character of any irreducible representation \rho as an integer linear combination \sum n_i \mathrm{Ind}_{H_i}^G \psi_i of characters induced from abelian characters \psi_i on subgroups H_i \leq G with n_i \in \mathbb{Z}.^[6] Since Artin L-functions multiply under tensor products and induction corresponds to such operations, it suffices to establish the functional equation for these induced characters. For abelian characters, global Artin reciprocity (from class field theory) identifies the L-function with a Hecke Grössencharacter L-function, which satisfies a known functional equation derived from local functional equations at each place via Tate's thesis.^[8]^[6] The local functional equations, holding for each place by explicit computation or local class field theory, combine via the global reciprocity to yield the overall equation.^[6]^[1]

Poles and Zeros

Artin L-functions L(s, \rho) for a representation \rho of the Galois group \mathrm{Gal}(K/k) admit meromorphic continuation to the entire complex plane, as established by Artin's original work and confirmed by Brauer's induction theorem decomposing representations into induced characters from cyclic subgroups. For the trivial representation \rho = 1, L(s, 1) = \zeta_K(s), the Dedekind zeta function of the number field K, which has a simple pole at s=1 with residue $2^{r_1} (2\pi)^{r_2} h_K R_K / (w_K \sqrt{|d_K|}), where r_1 is the number of real embeddings of K, r_2 the number of pairs of complex conjugate embeddings, h_K the class number, R_K the regulator, w_K the number of roots of unity, and d_K the discriminant. For irreducible non-trivial \rho, the functions are conjectured to be entire (holomorphic everywhere with no poles), per Artin's holomorphy conjecture; unconditionally, they are meromorphic, and any potential pole must be simple, located solely at s=1, and arise only from the multiplicity of the trivial representation within \rho. The order of vanishing of L(s, \rho) at s=0 is determined by the formula d_0(\rho) = \sum_{w \mid \infty} \dim \rho^{G_w} - \dim \rho^G, where the sum runs over the infinite places w of k, G_w denotes the decomposition group at w (generated by complex conjugation for real embeddings and trivial for complex ones), and G = \mathrm{Gal}(K/k). This quantity measures the excess dimension of the subspaces fixed by the local action at infinity over the global fixed space, capturing how the representation's structure under complex conjugation influences the local behavior at s=0; for example, if \rho is orthogonal (real type) with no global invariants, the order relates directly to the trace of the conjugation operator on the representation space. Non-vanishing results include the fact that L(s, \rho) has no zeros on the line \mathrm{Re}(s) = 1, a consequence of the Euler product and positivity of coefficients for \mathrm{Re}(s) > 1. The Generalized Riemann Hypothesis (GRH) posits that all non-trivial zeros lie on \mathrm{Re}(s) = 1/2, implying no zeros in \mathrm{Re}(s) > 1/2 and enabling effective error terms in Chebotarev density theorems. Artin L-functions satisfy the axioms of the Selberg class (Euler product, functional equation, polynomial growth of coefficients) except possibly the Ramanujan-Petersson conjecture on cusp forms, with GRH ensuring the critical line location of zeros. Partial results on zero-free regions exist for specific families; for instance, when \rho arises from symmetric powers of irreducible representations in low dimensions (e.g., degree 2 via modular forms), sieve-theoretic methods yield zero-free strips to the left of \mathrm{Re}(s) = 1 of width c / \log(|d_K| \log t) for some constant c > 0, improving classical bounds near s=1. In 2024, Lei, Vandermause, and Young established an approximate version of the holomorphy conjecture, showing that for irreducible Artin representations, any poles or essential singularities, if present, have bounded degree and multiplicity.^[9] Further progress on non-vanishing for \Re(s) > 1/2 appeared in 2025 work by Akbary and Hambrook.^[10] Numerical computations verify the GRH for zeros of small-degree Artin L-functions, such as those for dihedral representations of degree 3 or 4 over \mathbb{Q}, using Turing's method to count zeros up to height T with explicit bounds on discrepancies; for example, the first 100 zeros of L(s, \rho) for the Galois representation attached to the splitting field of x^3 + x^2 - 2x - 1 = 0 lie on the critical line within numerical precision.

Key Conjectures

Artin Conjecture

The Artin conjecture posits that for any finite-dimensional complex representation \rho of the Galois group \mathrm{Gal}(L/K) associated to a finite Galois extension L/K of number fields, the Artin L-function L(s, \rho) admits an analytic continuation to a holomorphic function on the entire complex plane, except in the case where \rho contains the trivial representation as a direct summand, in which event L(s, \rho) possesses a simple pole at s=1. This conjecture, formulated by Emil Artin in the 1920s, extends the known holomorphy properties of Dirichlet L-functions to the non-abelian setting, predicting no poles or essential singularities beyond the specified case.^[1]^[11] Partial results toward the conjecture have established holomorphy for representations of low dimension. For one-dimensional representations, which correspond to characters of abelian Galois groups, the conjecture reduces to the analytic continuation of Dirichlet L-functions, proven by Dirichlet in 1837 and extended to Hecke L-functions by Hecke in the 1930s.^[12] For two-dimensional representations over \mathbb{Q}, the conjecture has been established through the modularity theorem for Galois representations (Khare and Wintenberger, 2009).^[13] Additionally, for monomial representations—those induced from one-dimensional characters of subgroups—Brauer demonstrated meromorphy in 1947 using induction techniques, with poles controlled by the multiplicity of the trivial component.^[1] No counterexamples to the Artin conjecture are known, despite extensive computational checks for specific representations, such as icosahedral two-dimensional cases with small conductors.^[11] Significant modern progress traces to Robert Langlands' work in the 1970s, which reformulates the conjecture in terms of the correspondence between Galois representations and automorphic forms: if \rho lifts to a cuspidal automorphic representation on \mathrm{GL}_n(\mathbb{A}_K), then L(s, \rho) coincides with an automorphic L-function known to be entire.^[14] Local components of this functorial lift have been established for all dimensions via the local Langlands correspondence, but the global case remains open, with the full resolution anticipated from the Langlands reciprocity conjecture.^[1] If the Artin conjecture holds, the Artin L-functions would be primitive elements of the Selberg class of L-functions, serving as building blocks under the class's conjectured unique factorization.^[15]

Factorization of the Dedekind Zeta Function

For a finite abelian extension L/K of number fields, the Dedekind zeta function \zeta_L(s) factors as

\zeta_L(s) = \prod_{\chi} L(s, \chi),

where the product runs over all irreducible characters \chi of the abelian Galois group \Gal(L/K), and L(s, \chi) denotes the corresponding Artin L-function (which coincides with a Hecke L-function in this case). This equality holds in the region of absolute convergence \Re(s) > 1, and both sides admit meromorphic continuations to the complex plane that agree everywhere. This factorization resolves the holomorphy of \zeta_L(s)/\zeta_K(s) conjectured by Dedekind in the late 19th century.^[2] The result originated in efforts to generalize the Euler product for the Riemann zeta function to number fields, with Dedekind verifying special cases, such as pure cubic extensions. In the 1930s, Emil Artin established the factorization for cyclic extensions using his general reciprocity law, showing that it holds via explicit computations of local factors and densities. The full resolution for arbitrary abelian extensions followed from the development of class field theory, which identifies the characters with Hecke characters and confirms the meromorphic continuation of each factor. Independently, Richard Brauer provided a proof in 1947 applicable to all finite Galois extensions (including abelian ones) by demonstrating the meromorphy of general Artin L-functions through induction on the degree of the extension: any irreducible character decomposes as a non-negative integer linear combination of characters induced from cyclic subgroups, each of which yields a rational function times a product of Hecke L-functions. This factorization bears direct implications for the analytic class number formula in abelian extensions. The residue of \zeta_L(s) at s=1 equals $2^{r_1} (2\pi)^{r_2} h_L R_L / (w_L \sqrt{|\Delta_L|}), where h_L is the class number, R_L the regulator, w_L the number of roots of unity, r_1 the number of real embeddings, r_2 the number of pairs of complex embeddings, and \Delta_L the discriminant of L. Since \zeta_L(s) decomposes into the product over characters (with the trivial character contributing \zeta_K(s)), the residue factors correspondingly, yielding relations between the class number, regulator, and unit group of L and those of K; for instance, the relative class number h_L / h_K involves ratios of regulators R_L / R_K and discriminants, highlighting the arithmetic interplay in abelian towers.^[16] The result generalizes to non-abelian Galois extensions L/K via the analogous factorization \zeta_L(s) = \prod_{\rho} L(s, \rho)^{\dim \rho}, where \rho ranges over irreducible representations of \Gal(L/K); Brauer's theorem establishes this equality as meromorphic functions, though the Artin conjecture predicts additional holomorphy properties beyond the trivial representation. In the non-abelian setting, the factors may incorporate more intricate irrational elements in their special values compared to the rational Hecke L-functions of the abelian case. A representative example occurs for quadratic extensions K = \mathbb{Q}(\sqrt{d}) with fundamental discriminant d, where \Gal(K/\mathbb{Q}) \cong C_2 is cyclic of order 2. Here, the factorization simplifies to \zeta_K(s) = \zeta(s) L(s, \chi_d), with \chi_d the non-trivial character given by the Kronecker symbol ( \cdot / d ). This decomposition underpins the classical class number formula for quadratic fields, linking h_K explicitly to L(1, \chi_d).^[17]

Relations to Other L-Functions

Connection to Hecke L-Functions

Hecke L-functions arise in the context of number fields and are defined for Grössencharacters, which are characters ψ on the idèle class group of a number field K. Specifically, for such a ψ, the associated L-function is given by the Euler product

L(s, \psi) = \prod_{\mathfrak{p}} \det(I - \psi(\mathfrak{p}) N(\mathfrak{p})^{-s})^{-1},

where the product runs over the prime ideals of the ring of integers of K, and N(℘) denotes the norm of ℘. This definition captures the arithmetic data encoded by ψ, and Hecke established that these functions admit analytic continuation to the entire complex plane and satisfy a functional equation.^[1] In the special case where the Galois group Gal(K/k) is abelian for a Galois extension K/k of number fields, the Artin reciprocity map establishes an isomorphism between the Galois group and a quotient of the idèle class group of k. This identifies one-dimensional Galois characters ρ: Gal(K/k) → ℂ^× with Hecke characters ψ on the idèle class group of k, leading to the equality L(s, ρ) = L(s, ψ) up to finitely many Euler factors at ramified primes.^[6] Thus, Artin L-functions recover the classical Hecke L-functions precisely when the representation is one-dimensional and the extension is abelian.^[1] For higher-dimensional representations, Artin L-functions generalize this connection through induction and tensor products. If ρ is an irreducible representation of dimension greater than one, it can often be expressed as an induced representation from a subgroup or as a tensor product involving one-dimensional Hecke characters, allowing the Artin L-function to factor into products or quotients of Hecke L-functions via Brauer's induction theorem on characters. This decomposition facilitates the study of analytic properties by reducing to known Hecke factors.^[18] A prominent example occurs in cyclotomic fields, where the abelian Galois group over ℚ is isomorphic to units modulo the conductor, and the one-dimensional Artin characters correspond directly to Dirichlet characters. In this setting, the Artin L-functions coincide with the classical Dirichlet L-series, which are special cases of Hecke L-functions for the rationals. Another illustrative case involves elliptic curves with complex multiplication (CM) by an order in an imaginary quadratic field; the L-function of such a curve over a number field is expressible as a Hecke L-function attached to the Grössencharacter induced by the CM action, linking the arithmetic of the curve to modular forms whose L-functions align with these Hecke objects.^[6]^[18] While Hecke L-functions are always meromorphic on the complex plane with known poles only for the trivial character, Artin L-functions extend this framework to non-abelian representations but lack unconditional holomorphy; Artin's conjecture posits that they are entire except for the trivial representation, a statement proven in abelian cases via the Hecke theory but remaining open in general.^[1] This difference underscores how Artin L-functions broaden the abelian Hecke setting to capture non-abelian Galois data, albeit with conjectural analytic control.^[6]

Role in the Langlands Program

The Langlands reciprocity conjecture posits that Artin L-functions, attached to finite-dimensional representations of the Galois group of a number field, correspond to L-functions arising from automorphic representations of the general linear group GL_n over the adele ring of the field.^[19]^[20] Specifically, for an irreducible n-dimensional Galois representation ρ, the associated Artin L-function L(s, ρ) is expected to coincide with the automorphic L-function L(s, π_ρ) for a cuspidal automorphic representation π_ρ of GL_n(A_K), thereby establishing a deep reciprocity between Galois-theoretic and automorphic data.^[21] This correspondence forms a cornerstone of the Langlands program, bridging non-abelian Galois representations with the analytic properties of automorphic forms.^[22] Functoriality principles within the Langlands program further predict that lifts of Artin L-functions, such as those obtained via symmetric powers or exterior powers of the underlying Galois representation, yield automorphic L-functions on higher-rank GL_m.^[23] For instance, the conjecture asserts that the symmetric power Sym^k(ρ) corresponds to an automorphic representation on GL_{k+1}, which would imply the holomorphy of the original Artin L-function L(s, ρ) by transferring known analytic properties from the automorphic side.^[24] These functorial lifts provide a pathway to resolving the Artin conjecture on holomorphy through automorphic methods.^[1] Key advances include Langlands' development of base change and automorphic induction in the 1970s, which establish the automorphy of induced representations from cyclic extensions and prove holomorphy for associated Artin L-functions in low dimensions.^[22]^[25] In the 1980s, Clozel extended these techniques via base change for unitary groups and GL_n, demonstrating holomorphy for Artin L-functions attached to representations of nilpotent Galois groups.^[26] These results confirm the Langlands correspondence in specific cases, such as solvable images.^[27] Artin L-functions play a pivotal role in applications to the inverse Galois problem, where realizing prescribed Galois groups over the rationals often relies on embedding them into automorphic representations whose L-functions match Artin factors.^[1] Additionally, Artin representations connect to motive theory in the Langlands framework, where they are expected to underlie motives whose L-functions recover the Artin factors, facilitating comparisons between étale cohomology and automorphic forms.^[28] Recent progress includes the 2009 resolution of Serre's modularity conjecture by Khare and Wintenberger, which lifts two-dimensional Artin representations (modulo primes) to modular forms, thereby establishing their automorphy and holomorphy in this dimension.^[29]^[30]

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