nLab Artin L-function

Idea

An Artin L-function $L_\sigma$ (Artin 23) is an L-function associated with a number field $K$ and induced from the choice of an $n$-dimensional Galois representation, hence a linear representation

$$\sigma\;\colon\; Gal(L/K) \longrightarrow GL_n(\mathbb{C})$$

of the Galois group for some finite Galois extension $L$ of $K$: it is the product (“Euler product”) over all prime ideals $\mathfrak{p}$ in the ring of integers of $K$, of, essentially, the characteristic polynomials of the Frobenius homomorphism $Frob_p$ regarded (see here) as elements of Galois group

$$L_{K,\sigma} \colon s \mapsto \underset{\mathfrak{p}}{\prod} det \left( id - (N (\mathfrak{p}))^{-s} \sigma(Frob_{\mathfrak{p}}) \right)^{-1} \,$$

(e.g. Gelbhart 84, II.C.2, Snyder 02, def. 2.1.3).

discussion of ramified primes needs to be added

For $\sigma = 1$ the trivial representation then the Artin L-function reduces to the Dedekind zeta function (see below). So conversely one may think of Artin L-functions as being Dedekind zeta functions which are “twisted” by a Galois representation. (Notice that Galois representations are the analog in arithmetic geometry of flat connections/local systems of coefficients).

For $\sigma$ any 1-dimensional Galois representation (hence the case $n = 1$) then there is a Dirichlet character $\chi$ such that the Artin L-function $L_\sigma$ is equal to the Dirichlet L-function $L_\chi$ – this relation is part of Artin reciprocity.

For $\sigma$ any $n$-dimensional representation for $n \geq 1$ then the conjecture of Langlands correspondence is that for each $n$-dimensional Galois representation $\sigma$ there is an automorphic representation $\pi$ such that the Artin L-function $L_\sigma$ equals the automorphic L-function $L_\pi$ (e.g Gelbhart 84, pages 5-6).

Properties

For irreducible representations – Artin’s conjecture

Artin’s conjecture is the statement that for a nontrivial irreducible representation $\sigma$ the Artin L-function $L_{K,\sigma}$ is not just a meromorphic function on the complex plane, but in fact an entire holomorphic function.

e.g. (Ram Murty 94, p. 3)

or rather with at most a pole at $s = 1$ Murty-Murty 12, page 29 in chapter 2

For induced representations

Let $H \hookrightarrow Gal(L/K)$ be subgroup of the Galois group $G \coloneqq Gal(L/K)$ and write $L^H \hookrightarrow L$ for the subfield of elements fixed by $H$. Let $\sigma$ be a representation of $H = Gal(L/L^H)$ and write $Ind_H^G\sigma$ for the induced representation of $G$. Then the corresponding Artin L-functions are equal:

$$L_{K,{Ind_H^{G}\sigma}} = L_{L^H, \sigma} \,.$$

(e.g. (Murty-Murty 12, equation (2) in chapter 2)).

Relation to the Dedekind zeta function

For $\sigma = 1$ the trivial representation then $\sigma(Frob_{\mathfrak{p}}) = id$ identically, and hence in this case the definition of the Artin L-function becomes verbatim that of the Dedekind zeta function $\zeta_K$:

$$L_{L,1} = \zeta_L \,.$$

If $L/K$ is a Galois extension, the by the behaviour of Artin L-functions for induced representation as above this is also the Artin L-function of $K$ itself for the regular representation of $Gal(L/K)$

$$\zeta_L = L_{L,1} = L_{K,{Ind_1^{Gal(L/K)}1}}$$

(e.g. (Murty-Murty 12, below (2) in chapter 2))

Analogy with Selberg/Ruelle zeta-functions

The Frobenius morphism $Frob_p$ giving an element in the Galois group means that one may think of it as an element of the fundamental group of the given arithmetic curve (see at algebraic fundamental group). There is a direct analogy between Frobenius elements at prime numbers in arithmetic geometry and parallel transport along prime geodesics in hyperbolic geometry (Brown 09, p. 6).

Under this interpretation, a Galois connection corresponds to a flat connection (local system of coefficients) on an arithmetic curve, and its Artin L-function is a product of characteristic polynomials of the monodromies/holonomies of that flat connection.

Now, in differential geometry, given a suitable odd-dimensional hyperbolic manifold equipped with an actual flat bundle over it, then associated with it is the Selberg zeta function and Ruelle zeta function. Both are (by definition in the latter case and by theorems in the former) Euler products of characteristic polynomials of monodromies/holonomies. See at Selberg zeta function – Analogy with Artin L-function and at Ruelle zeta function – Analogy with Artin L-function for more on this.

See also (Brown 09, page 6, Morishita 12, remark 12.7).

(The definition also has some similarity to that of the Alexander polynomial, see at arithmetic topology.)

Related concepts

context/function field analogy	theta function $\theta$	zeta function $\zeta$ (= Mellin transform of $\theta(0,-)$)	L-function $L_{\mathbf{z}}$ (= Mellin transform of $\theta(\mathbf{z},-)$)	eta function $\eta$	special values of L-functions
physics/2d CFT	partition function $\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2))$ as function of complex structure $\mathbf{\tau}$ of worldsheet $\Sigma$ (hence polarization of phase space) and background gauge field/source $\mathbf{z}$	analytically continued trace of Feynman propagator $\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tau$	analytically continued trace of Feynman propagator in background gauge field $\mathbf{z}$: $L_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tau$	analytically continued trace of Dirac propagator in background gauge field $\mathbf{z}$ $\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s$	regularized 1-loop vacuum amplitude $pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)$ / regularized fermionic 1-loop vacuum amplitude $pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right)$ / vacuum energy $-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)$
Riemannian geometry (analysis)		zeta function of an elliptic differential operator	zeta function of an elliptic differential operator	eta function of a self-adjoint operator	functional determinant, analytic torsion
complex analytic geometry	section $\theta(\mathbf{z},\mathbf{\tau})$ of line bundle over Jacobian variety $J(\Sigma_{\mathbf{\tau}})$ in terms of covering coordinates $\mathbf{z}$ on $\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})$	zeta function of a Riemann surface	Selberg zeta function		Dedekind eta function
arithmetic geometry for a function field		Goss zeta function (for arithmetic curves) and Weil zeta function (in higher dimensional arithmetic geometry)
arithmetic geometry for a number field	Hecke theta function, automorphic form	Dedekind zeta function (being the Artin L-function $L_{\mathbf{z}}$ for $\mathbf{z} = 0$ the trivial Galois representation)	Artin L-function $L_{\mathbf{z}}$ of a Galois representation $\mathbf{z}$, expressible “in coordinates” (by Artin reciprocity) as a finite-order Hecke L-function (for 1-dimensional representations) and generally (via Langlands correspondence) by an automorphic L-function (for higher dimensional reps)		class number $\cdot$ regulator
arithmetic geometry for $\mathbb{Q}$	Jacobi theta function ($\mathbf{z} = 0$)/ Dirichlet theta function ($\mathbf{z} = \chi$ a Dirichlet character)	Riemann zeta function (being the Dirichlet L-function $L_{\mathbf{z}}$ for Dirichlet character $\mathbf{z} = 0$)	Artin L-function of a Galois representation $\mathbf{z}$ , expressible “in coordinates” (via Artin reciprocity) as a Dirichlet L-function (for 1-dimensional Galois representations) and generally (via Langlands correspondence) as an automorphic L-function

References

The original article is

• Emil Artin, Über eine neue Art von L Reihen. Hamb. Math. Abh. 3. (1923) Reprinted in his collected works, ISBN 0-387-90686-X. English translation in (Snyder 02, section A)

Reviews include

• Wikipedia, Artin L-function

• M. Ram Murty, V. Kumar Murty, Non-vanishing of L-functions and applications, Modern Birkhäuser classics 2012 (chapter 2 pdf)

• Noah Snyder, Artin L-Functions: A Historical Approach, 2002 (pdf)

and in the context of the Langlands program

• Stephen Gelbart, An elementary introduction to the Langlands program, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219 (web)

Further development includes

• M. Ram Murty, Selberg’s conjectures and Artin -functions, Bull. Amer. Math. Soc. 31 (1994), 1-14 (web)

The analogy with the Selberg zeta function is discussed in

• Darin Brown, Lifting properties of prime geodesics, Rocky Mountain J. Math. Volume 39, Number 2 (2009), 437-454 (euclid)

• Masanori Morishita, section 12.1 of Knots and Primes: An Introduction to Arithmetic Topology, 2012 (web)

The analogies between Alexander polynomial and L-functions and touched upon in

• Ken-ichi Sugiyama, The properties of an L-function from a geometric point of view, 2007 pdf; A topological $\mathrm{L}$ -function for a threefold, 2004 pdf