Artin L-function

Properties

Most of the properties of Artin L-functions generalize properties known for other functions like Dedekind zeta functions and Dirichlet L-functions.

Basic properties

Additivity

Artin L-functions behave well with respect to direct sum of representations:

${\displaystyle L(s,{\rho }_1\oplus {\rho }_2,L/K)=L(s,{\rho }_1,L/K)\cdot L(s,{\rho }_2,L/K)}$

Inductivity

Artin L-functions behave well to induction of representations. Let ${\displaystyle K\subseteq L\subseteq M}$ be tower of finite number field extensions. Let ${\displaystyle H=\text{Gal}(M/L)}$ and ${\displaystyle G=\text{Gal}(M/K)}$ be Galois groups for respective extensions. It is easy to observe that ${\displaystyle H}$ is subgroup of ${\displaystyle G}$. If ${\displaystyle \rho }$ is representation of ${\displaystyle H}$, then for induced representation on larger group:

${\displaystyle L(s,\rho ,M/L)=L(s,{\text{Ind}}_H^G(\rho ),M/K)}$

Inflation

It is easy to check by comparing Euler products over prime ideals, that for the trivial representation of the Galois group, the Artin L-function is the Dedekind zeta function of the smaller field:

${\displaystyle L(s,\mathfrak{1},L/K)={\zeta }_K(s)}$

Since this function is independent of the larger field, this property is called inflation.

Relation to characters

Denote the character of the representation ${\displaystyle \rho }$ as ${\displaystyle {\chi }_{\rho }}$. From the formula:

${\displaystyle \log (det(M))=\mathrm{tr}(\log (M))}$

it follows that logarithm of every factor of Artin L-function can be written as:

${\displaystyle -\log (det([\mathrm{Id}-N(\mathfrak{p}{)}^{-s}\cdot \rho ({\mathbf{F}\mathbf{r}\mathbf{o}\mathbf{b}}_{\mathfrak{p}})]{|}_{V^{I_{\mathfrak{p}}}}))=\sum _{m=1}^{\mathrm{\infty }}\frac{{\chi }_{\rho {|}_{V^{I_{\mathfrak{p}}}}}({\mathbf{F}\mathbf{r}\mathbf{o}\mathbf{b}}_{\mathfrak{p}}^m)}{m}N(\mathfrak{p}{)}^{-sm}}$

Taking the logarithm of the product of all factors of the function:

${\displaystyle \log (L(s,\rho ,L/K))=-\sum _{\mathfrak{p}\subseteq {\mathcal{O}}_K}\sum _{m=1}^{\mathrm{\infty }}\frac{{\chi }_{\rho {|}_{V^{I_{\mathfrak{p}}}}}({\mathbf{F}\mathbf{r}\mathbf{o}\mathbf{b}}_{\mathfrak{p}}^m)}{m}N(\mathfrak{p}{)}^{-sm}}$

From this relation it follows that for a given extension of number fields, the Artin L-function can be defined using only the character of the representation. Particularly, equivalent representations define the same Artin L-function. Since the character is a class function, this shows that after restriction to ${\displaystyle V^{I_{\mathfrak{p}}}}$ value of factor is independent of the choice of Frobenius element.

Each basic property of Artin L-function can also be rewritten in terms of characters. If ${\displaystyle {\chi }_1,{\chi }_2}$ are characters of respectively ${\displaystyle {\rho }_1,{\rho }_2}$ then additivity property becomes:

${\displaystyle L(s,{\chi }_1+{\chi }_2,L/K)=L(s,{\chi }_1,L/K)\cdot L(s,{\chi }_2,L/K)}$

If ${\displaystyle {\chi }_{\mathfrak{1}}}$ is character of trivial representation, then inflation property is equivalent to:

${\displaystyle L(s,{\chi }_{\mathfrak{1}},L/K)={\zeta }_K(s)}$

If ${\displaystyle {\chi }_H^G}$ is induced character then inductivity property is equivalent to:

${\displaystyle L(s,\chi ,M/L)=L(s,{\chi }_H^G,M/K)}$

Character form of L-function is useful for showing some properties like existence of analytic continuation or factorization of Dedekind zeta functions by Artin L-functions.

Factorization of Dedekind zeta functions

One of the main applications of Artin L-functions is to provide non-trivial factorizations for Dedekind zeta functions.

Galois case

If ${\displaystyle L/K}$ is a Galois extension, then for the regular representation of the Galois group, the Artin L-function is the Dedekind zeta function of the larger field:

${\displaystyle L(s,{\rho }_{\text{reg}},L/K)={\zeta }_L(s)}$

Since Artin L-functions behave well with respect to direct sums and regular representations for finite groups decompose into irreducible representations, it is usually written as

${\displaystyle {\zeta }_L(s)=\prod _{\rho \text{ irr}}L(s,\rho ,L/K{)}^{\mathrm{deg}(\rho )}}$

where the index runs over all irreducible representations of Galois group. The proof of this equality is rather straightforward, but some steps require results from character theory and splitting of prime ideals in Galois extensions.

General case

It is possible to obtain the general formula for factorization of Dedekind zeta function for a larger field, but it become more complicated and relies on inductivity property. Denote:

• ${\displaystyle M}$ be for normal closure for ${\displaystyle L/K}$
• ${\displaystyle H}$ be Galois group for ${\displaystyle M/L}$
• ${\displaystyle G}$ be Galois group for ${\displaystyle M/K}$
• ${\displaystyle ⟨\cdot ,\cdot ⟩}$ be product on characters
• ${\displaystyle {\chi }_{\rho }}$ be character of irreducible representation
• ${\displaystyle {\chi }^{\ast }}$ be character of ${\displaystyle {\text{Ind}}_H^G(\mathcal{1})}$

Then the factorization of Dedekind zeta function is as follows:

${\displaystyle {\zeta }_L(s)=\prod _{\rho \text{ irr}}L(s,{\text{Ind}}_H^G(\rho ),M/K{)}^{⟨{\chi }_{\rho },{\chi }^{\ast }⟩}}$

Analytic continuation

In the general case Artin L-functions were proven to have meromorphic continuation to complex plane. Using the Brauer theorem on induced characters it can be shown that each character can be written as:

${\displaystyle {\chi }_{\rho }=\sum _{i=1}^k{m}_i\cdot {\chi }_i}$

where ${\displaystyle m_i}$ are integers and ${\displaystyle {\chi }_i}$ are characters induced from 1-dimensional representations of subgroups. Using inductivity, additivity and Artin reciprocity, the function can be written as:

${\displaystyle L(s,{\chi }_{\rho },L/K)=\prod _{i=1}^{k}L(s,{\chi }_i{)}^{m_i}}$

where in product we have L-functions for corresponding Hecke characters. Since Hecke L-functions are meromorphic (with only possible pole at ${\displaystyle s=1}$, when Hecke character is trivial), it gives existence of meromorphic continuation to complex plane.

Brauer's result is already improvement of original theorem proven by Artin, which guaranteed only that some power of Artin L-function is meromorphic, since Artin's theorem used rational coefficients instead of integer.

Functional equation

Artin L-functions satisfy a functional equation involving Euler gamma function that relates values of ${\displaystyle L(s,\rho ,L/K)}$ with ${\displaystyle L(1-s,{\rho }^{\ast },L/K)}$, where ${\displaystyle {\rho }^{\ast }}$ denotes the complex conjugate representation. Each gamma factor in this equation is tied to Archimedean places introduced on field, in a similar way as factors in Euler product are tied to p-adic places.

Using modern notation from the theory of automorphic forms, let's denote Archimedean gamma factors:

${\displaystyle {\mathrm{\Gamma }}_{\mathbb{R}}(s)={\pi }^{-\frac{s}{2}}\mathrm{\Gamma }(\frac{s}{2}){\mathrm{\Gamma }}_{\mathbb{C}}(s)=2(2\pi {)}^{-s}\mathrm{\Gamma }(s)}$

We define numbers that will be used as exponents in factors of functional equation. For real places, choose a generator ${\displaystyle {\varphi }_v}$ of group ${\displaystyle \text{Gal}(L_v/K_v)}$. Then ${\displaystyle \rho ({\varphi }_v)}$ is an involution on ${\displaystyle V}$ and has two eigenspaces related to eigenvalues ${\displaystyle 1,-1}$:

${\displaystyle n_{+}=\dim V_{+}{n}_{-}=\dim V_{-}}$

Direct computation shows that alternatively this number can be defined using character of representation:

${\displaystyle n_{+}=\frac{{\chi }_{\rho }(1)+{\chi }_{\rho }({\varphi }_v)}{2},n_{-}=\frac{{\chi }_{\rho }(1)-{\chi }_{\rho }({\varphi }_v)}{2}}$

The factors for Archimedean places depends on whether the place is real or complex:

${\displaystyle L_v(s,\rho ,L/K)=\{\begin{array}{ll}{\mathrm{\Gamma }}_{\mathbb{R}}(s{)}^{n_{+}}{\mathrm{\Gamma }}_{\mathbb{R}}(s+1{)}^{n_{-}}& v\text{ real}\\ {\mathrm{\Gamma }}_{\mathbb{C}}(s{)}^{\dim \rho }& v\text{ complex}\end{array}}$

The conductor that encodes information about ramification is defined in a following way:

${\displaystyle c(L/K,\rho )={\mathrm{\Delta }}_K\cdot N(\mathfrak{f}(\rho ))}$

where ${\displaystyle {\mathrm{\Delta }}_K}$ is discriminant of ${\displaystyle K}$ and ${\displaystyle N(\mathfrak{f}(\rho ))}$ is absolute norm of Artin conductor for representation ${\displaystyle \rho }$. Now, define the completed L-function as the product of factors for all places on ${\displaystyle L/K}$, including the Archimedean ones and conductor for that field extension and representation:

${\displaystyle \mathrm{\Lambda }(s,\rho ,L/K)=c(L/K,\rho {)}^{\frac{s}{2}}\prod _v{L}_v(s,\rho ,L/K)}$

The "completed" Artin L-function satisfies the following functional equation (with conjugated representation):

${\displaystyle \mathrm{\Lambda }(s,\rho ,L/K)=ϵ(\rho ,L/K)\mathrm{\Lambda }(1-s,{\rho }^{\ast },L/K)}$,

where ${\displaystyle ϵ(\rho )}$ is some complex number of absolute value 1, called the Artin root number. This functional equation generalizes equations for Hecke L-functions and Dedekind zeta functions, especially archetypical equation for Riemann zeta function. The case when this equation relates L-function to itself is exactly when ρ and ρ* are equivalent representations. It is, algebraically speaking, the case when ρ is a real representation or quaternionic representation.

The Artin root number is the subject of significant research. Robert Langlands and Pierre Deligne established a factorisation of Artin root number into Langlands–Deligne local constants; this is significant in conjectural relationships to automorphic representations. The Artin root number is, then, either +1 or −1. The question of which sign occurs is linked to Galois module theory.

The functional equation allows one to distinguish between trivial and nontrivial zeros of Artin L-function. Zeros that cancels the poles of Gamma functions are called trivial zeros. Functional equation guarantees that nontrivial zeros and possible poles (conjectured to not exist) must lie in critical strip: ${\displaystyle 0\le \mathrm{Re}(s)\le 1}$ and are symmetric with respect to critical line ${\displaystyle \mathrm{Re}(s)=\frac{1}{2}}$.

Artin reciprocity

A key result in class field theory, Artin reciprocity usually is expressed as existence of certain homomorphism between idele class group and abelianization of Galois group. Equivalent statement says that if ${\displaystyle L/K}$ is abelian extension and ${\displaystyle \rho }$ is irreducible representation of its Galois group, then there exists Hecke character ${\displaystyle {\chi }_K}$ such that:

${\displaystyle L(s,\rho ,L/K)=L(s,{\chi }_K)}$

If we have ${\displaystyle K=\mathbb{Q}}$, then Hecke character can be replaced by Dirichlet character and Hecke L-function becomes Dirichlet L-function. Combining Artin reciprocity with factorizations of Dedekind zeta functions, we would get that for abelian extension:

${\displaystyle {\zeta }_L(s)=\prod _{\chi \text{ prim}}L(s,{\chi }_K)}$

Where index runs over primitive Hecke characters of ${\displaystyle K}$, corresponding to irreducible representations of ${\displaystyle \text{Gal}(L/K)}$. In even more special case, if ${\displaystyle L/\mathbb{Q}}$ is abelian extension, then:

${\displaystyle {\zeta }_L(s)=\prod _{\chi \text{ prim}}L(s,\chi )}$

where ${\displaystyle L(\chi ,s)}$ are Dirichlet L-functions for primitive Dirichlet characters corresponding to irreducible representations of ${\displaystyle \text{Gal}(L/\mathbb{Q})}$.

Conjectures

Some properties of Artin L-functions have turned out to become very difficult to be proven. Many of them are generalizations of properties displayed by special cases of them and have vital meaning for number theory.

The Artin conjecture

The Artin conjecture also called Artin holomorphy conjecture states that ${\displaystyle L(s,\rho ,L/K)}$ for a non-trivial, irreducible representation ${\displaystyle \rho }$ is entire function..

Assumption that representation is non-trivial is necessary, because for trivial representation we get Dedekind zeta function with one simple pole at ${\displaystyle s=1}$. Assumption that representation is irreducible allows to simplify statement excluding representation induced from trivial representation. However this statement is natural generalization of theorem that Hecke L-functions are entire for non-trivial, primitive Hecke character.

Main problem with holomorphy of analytic continuation arises from fact that meromorphic continuation is constructed as product of Hecke L-functions in integer powers: zeros of functions used with negative powers produce poles that must be cancelled by zeros of functions used with positive powers. Direct proof of Artin conjecture from this construction would require very detailed knowledge about location and multiplicities of nontrivial zeros of Hecke L-functions, problem that is generally very hard for L-functions.

Further improvement of Brauer's theorem to use only positive integers coefficient is not possible in general case: groups where all characters can be induced from one-dimensional characters using possitive integer coefficients are called monomial groups, and Taketa (1930) proved that all monomial group are solvable groups. Moreover, this is only necessary condition: the SL(2,3) is solvable group that has complex character of 2-dimensional representation that cannot be expressed this way.

Artin conjecture was solved in very specific cases:

• for one-dimensional representations conjecture follows from holomorphy of Hecke L-functions. More generally Artin, showed that the conjecture is true for all representations induced from 1-dimensional representations.
• in case of two-dimensional representations, they can be classified by the nature of the image subgroup: it may be cyclic, dihedral, tetrahedral, octahedral, or icosahedral.
  • For odd and irreducible, two-dimensional representations follows from the proof of Serre's modularity conjecture, regardless of image subgroup.
  • For the cyclic or dihedral case follows easily from Erich Hecke's work.
  • Langlands (1980) used the base change lifting to prove the tetrahedral case.
  • Tunnell (1981) extended Langlands work to cover the octahedral case.
  • The general icosahedral case is open and is active area of research.

Artin conjecture combined with factorization of Dedekind zeta function for Galois extensions implies Dedekind conjecture, that for any finite extension ${\displaystyle L/K}$ of number fields the quotient:

${\displaystyle \frac{{\zeta }_L(s)}{{\zeta }_K(s)}}$

is entire function.

Langlands reciprocity

Langlands reciprocity is conjecture proposed by Robert Langlands as a generalization of Artin reciprocity to non-abelian field extensions.

Let ${\displaystyle L/K}$ be finite number field extension and ${\displaystyle \rho }$ be irreducible representation of ${\displaystyle \text{Gal}(L/K)}$. Langlands conjectured that, then exists automorphic cuspidal representation ${\displaystyle \pi }$ of general linear group ${\displaystyle \text{GL}(n,A_K)}$, where ${\displaystyle {\mathbf{A}}_K}$ be adele ring for ${\displaystyle K}$ satisfying:

${\displaystyle L(s,\rho ,L/K)=L(s,\pi )}$

where ${\displaystyle L(s,\pi )}$ is automorphic L-function for that representation. This reciprocity law was proven in very special cases and is still largely conjectural, but stands for the best proposition for development of non-abelian class field theory and solution to Hilbert's ninth problem.

Langlands (1970) pointed out that the Artin conjecture would follow from this reciprocity law and conjecture that the L-functions of cuspidal automorphic representations are entire functions. This was one of the major motivations for more much general Langlands program.

The Riemann Hypothesis

From functional equation, all nontrivial zeros for Artin L-functions are symmetric with respect to critical line ${\displaystyle \mathrm{Re}(s)=\frac{1}{2}}$. Riemann Hypothesis for Artin L-functions says, that they should lie on critical line. This is generalization of original Riemann hypothesis as well as its versions for Dirichlet L-functions, Hecke L-functions and Dedekind zeta functions, then it have many consequences in analytic and algebraic number theory.

Assuming Artin's conjecture it is equivalent with Riemann Hypotheses for Hecke L-functions and Dedekind zeta functions. Assuming Langlands reciprocity, it would follow from Grand Riemann hypothesis.

Stark conjectures

Using class number formula and functional equation for Dedekind zeta function we would get that order and leading coefficient of trivial zero at ${\displaystyle s=0}$ is related to some important invariants of number field:

${\displaystyle \underset{s\to 0}{lim}{s}^{-(r_1+r_2-1)}{\zeta }_K(s)=-\frac{h(K)R(K)}{w(K)}}$

where:

• ${\displaystyle h(K)}$ is class number for ${\displaystyle K}$
• ${\displaystyle R(K)}$ is regulator for ${\displaystyle K}$
• ${\displaystyle w(K)}$ is number of roots of unity in ${\displaystyle K}$
• ${\displaystyle r_1}$ is number of real places of ${\displaystyle K}$
• ${\displaystyle r_2}$ is number of non-conjugated complex places of ${\displaystyle K}$
• ${\displaystyle r_1+r_2-1}$ is both order of zero as rank of unit group in ${\displaystyle {\mathcal{O}}_K}$

As a generalization of that phenomenon Stark conjectured that leading coefficient of Taylor series of Artin L-function at ${\displaystyle s=0}$ should be product of Stark regulator with some algebraic number. Stark proposed refined version of his conjecture when ${\displaystyle L/K}$ is abelian extension and order of zero of Artin L-function at ${\displaystyle s=0}$ is one. He predicted existence of Stark units whose roots should generate Kummer extensions of ${\displaystyle K}$ and having implications for Hilbert's twelfth problem.