Artin L-function

Remarks

• Almost all prime ideals in finite extension of number field are unramified, in their case ${\displaystyle I_{\mathfrak {p}}}$ is trivial group and ${\displaystyle V^{I_{\mathfrak {p}}}=V}$. Restriction of linear mapping is used to deal with ramified prime ideals which usually appears in number field extensions and are related to vital information about them.
• Frobenius element don't need to be unique. For unramified primes all Frobenius elements are conjugated in Galois group, hence value of factor is the same for all possible choices (see Properties below). For ramified primes, except conjugation, Frobenius element can be each element of coset with respect to ${\displaystyle I_{\mathfrak {p}}}$, thus restricting of linear mappings to ${\displaystyle V^{I_{\mathfrak {p}}}}$ makes value unique.
• It is arguably more correct to think instead about the coinvariants, the largest quotient space fixed by ${\displaystyle I_{\mathfrak {p}}}$, rather than the invariants, but the result here will be the same.

Basic properties

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 Galois extension, then for regular representation of Galois group Artin L-function is 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 usualy written as:

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

Where 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.

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Proof

Let ${\displaystyle n=[L:K]}$. By ${\displaystyle L_{\mathfrak {p}}(s,\rho ,L/K)}$ denote factor of L-function for prime ideal ${\displaystyle {\mathfrak {p}}}$ in ${\displaystyle {\mathcal {O}}_{K}}$ and by ${\displaystyle \chi _{\rho }}$ character of representation restricted to ${\displaystyle V^{I_{\mathfrak {p}}}}$. For ${\displaystyle {\mathfrak {p}}}$ we have factorization: ${\displaystyle {\mathfrak {p}}{\mathcal {O}}_{L}=\prod _{i=1}^{g}P_{i}^{e_{i}}}$ where ${\displaystyle P_{i}}$ are prime ideals lying above ${\displaystyle {\mathfrak {p}}}$ in this extension with inertia degrees ${\displaystyle f_{i}}$. Using that ${\displaystyle L/K}$ is Galois extension, then for each ${\displaystyle i,j=1,\dots ,g}$ there is: ${\displaystyle e_{i}=e_{j}\quad {\text{,}}\quad f_{i}=f_{j}\quad {\text{and}}\quad efg=n}$ ${\displaystyle e=e_{i}}$ is called ramification index and ${\displaystyle f=f_{i}}$ is called inertia degree for ideal ${\displaystyle {\mathfrak {p}}}$, and there is: ${\displaystyle }$. Now take logarithm of factor of L-function for this ideal: ${\displaystyle -\log(L_{\mathfrak {p}}(s,\rho ,L/K))=\sum _{m=1}^{\infty }{\frac {\chi _{\rho }(\mathbf {Frob} _{\mathfrak {p}}^{m})}{m}}N({\mathfrak {p}})^{-sm}}$ Before summation for all irreducible representations recall fact from character theory, that if index runs all irreducible representations of finite group ${\displaystyle G}$: ${\displaystyle \sum _{\rho {\text{ irr}}}\chi _{\rho }(e_{G})\chi _{\rho }(\sigma ))={\begin{cases}|G|&\sigma =e_{G}\\0&\sigma \neq e_{G}\end{cases}}}$ where ${\displaystyle e_{G}}$ denotes neutral element of group ${\displaystyle G}$ (don't confuse with ramification index). After restriction to subspace ${\displaystyle V^{I_{\mathfrak {p}}}}$ irreducible representations of ${\displaystyle G}$ actually corresponds to irreducible representations of quatient group. Since ramification index equals order of inertia group, and for Galois extension order of Galois group equals degree of extension: ${\displaystyle |G|=|{\text{Gal}}(L/K)/I_{\mathfrak {p}}|={\tfrac {n}{e}}}$. Taking summation for all irreducible representations and using fact that order of Frobenius element in ${\displaystyle {\text{Gal}}(L/K)/I_{\mathfrak {p}}}$ is equal to inertia degree, we will get: ${\displaystyle \sum _{\rho {\text{ irr}}}\sum _{m=1}^{\infty }{\frac {\chi (e_{G})\chi _{\rho }(\mathbf {Frob} _{\mathfrak {p}}^{m})}{m}}N({\mathfrak {p}})^{-sm}={\frac {n}{e}}\sum _{m=1}^{\infty }{\frac {1}{fm}}N({\mathfrak {p}})^{-sfm}}$ Using that ${\displaystyle efg=n}$, we will get: ${\displaystyle {\frac {n}{e}}\sum _{m=1}^{\infty }{\frac {1}{fm}}N({\mathfrak {p}})^{-sfm}=g\sum _{m=1}^{\infty }{\frac {1}{m}}N({\mathfrak {p}})^{-sfm}}$ Using formula for Taylor expansion of logarithm we can collapse series at right side: ${\displaystyle g\sum _{m=1}^{\infty }{\frac {1}{m}}N({\mathfrak {p}})^{-sfm}=-g\log(1-N({\mathfrak {p}})^{-sf})}$ Exponentiating both sides and using ${\displaystyle \chi (e_{G})=\deg \rho }$ we get: ${\displaystyle \prod _{\rho {\text{ irr}}}L_{\mathfrak {p}}(s,\rho ,L/K)^{\deg(\rho )}={\frac {1}{(1-N({\mathfrak {p}})^{-fs})^{g}}}}$ Now using fact that there are ${\displaystyle g}$ distinct prime ideals over ${\displaystyle {\mathfrak {p}}}$, for each of them their absolute norms: ${\displaystyle N(P_{i})=N(P_{j})}$ and that: ${\displaystyle N({\mathfrak {p}})^{f}=N(P_{i})}$ we will get: ${\displaystyle {\frac {1}{(1-N({\mathfrak {p}})^{-fs})^{g}}}=\prod _{i=1}^{g}{\frac {1}{1-N(P_{i})^{-s}}}}$ Finally taking product over all prime ideals in ${\displaystyle {\mathcal {O}}_{K}}$ and taking notice on splitting of ideals we would get: ${\displaystyle \prod _{\rho {\text{ irr}}}L(s,\rho ,L/K)^{\operatorname {deg} (\rho )}=\prod _{{\mathfrak {p}}\subseteq {\mathcal {O}}_{K}}{\frac {1}{(1-N({\mathfrak {p}})^{-fs})^{g}}}=\prod _{P\subseteq {\mathcal {O}}_{L}}{\frac {1}{1-N(P)^{-s}}}}$ Which equals Euler product for Dedekind zeta function for ${\displaystyle L}$: ${\displaystyle \prod _{P\subseteq {\mathcal {O}}_{L}}{\frac {1}{1-N(P)^{-s}}}=\zeta _{L}(s)}$

Close

Analytic continuation

In general case Artin L-functions were proven to have meromorphic continuation to complex plane. Using 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 it gives that 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 ${\displaystyle L(1-s,\rho ^{*},L/K)}$, where ${\displaystyle \rho ^{*}}$ denotes the complex conjugate representation. Each gamma factor in this equation is closely 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, denote basic factors:

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

Define numbers that will be used as exponents in factors of functional equation. For real place let us choose generator ${\displaystyle \phi _{v}}$ of group ${\displaystyle {\text{Gal}}(L_{v}/K_{v})}$. Then ${\displaystyle \rho (\phi _{v})}$ is involution on ${\displaystyle V}$ and have two eigenspaces related to eigenvalues: ${\displaystyle 1,-1}$.

${\displaystyle n_{+}=\dim V_{+}\qquad n_{-}=\dim V_{-}}$

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

${\displaystyle n_{+}={\frac {\chi _{\rho }(1)+\chi _{\rho }(\phi _{v})}{2}},\quad n_{-}={\frac {\chi _{\rho }(1)-\chi _{\rho }(\phi _{v})}{2}}}$

Factor for archimedean place is dependent on whether the place is real or complex:

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

Define function being product of factors for all places on ${\displaystyle L/K}$, including archimedean ones:

${\displaystyle \Lambda (s,\rho ,L/K)=\prod _{v}L_{v}(s,\rho ,L/K)}$

Then "completed" Artin L-function satisfies the following functional equation with function for conjugated representation:

${\displaystyle \Lambda (s,\rho ,L/K)=\epsilon (\rho ,L/K)\Lambda (\rho ^{*},1-s,L/K)}$,

where ${\displaystyle \epsilon (\rho )}$ is some complex number of absolute value 1, called 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.

Artin root number is the subject of research itself. 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.^[1]

Functional equation allows 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: ${\displaystyle 0\leq \operatorname {Re} (s)\leq 1}$ and are symmetric with respect to critical line ${\displaystyle \operatorname {Re} (s)={\tfrac {1}{2}}}$.

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 ${\displaystyle {\text{Gal}}(L/\mathbb {Q} )}$.

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.^[2].

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 ${\displaystyle \operatorname {Re} (s)={\tfrac {1}{2}}}$. 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:

${\displaystyle \lim _{s\rightarrow 0}s^{-(r_{1}+r_{2}-1)}\zeta _{K}(s)=-{\frac {h(K)R(K)}{w(K)}}}$

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.