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\begin{document}

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\section*{Artin L-function}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{theta_functions}{}\paragraph*{{Theta functions}}\label{theta_functions}

[[!include theta functions - contents]]

\hypertarget{arithmetic_geometry}{}\paragraph*{{Arithmetic geometry}}\label{arithmetic_geometry}

[[!include arithmetic geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{for_irreducible_representations__artins_conjecture}{For irreducible representations -- Artin's conjecture}\dotfill \pageref*{for_irreducible_representations__artins_conjecture} \linebreak
\noindent\hyperlink{ForInducedRepresentations}{For induced representations}\dotfill \pageref*{ForInducedRepresentations} \linebreak
\noindent\hyperlink{RelationToDedekindZeta}{Relation to the Dedekind zeta function}\dotfill \pageref*{RelationToDedekindZeta} \linebreak
\noindent\hyperlink{AnalogyWithSelbergZeta}{Analogy with Selberg/Ruelle zeta-functions}\dotfill \pageref*{AnalogyWithSelbergZeta} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

An \emph{Artin L-function} $L_\sigma$ (\hyperlink{Artin23}{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]]

\begin{displaymath}
\sigma\;\colon\; Gal(L/K) \longrightarrow GL_n(\mathbb{C})
\end{displaymath}
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 \href{Frobenius+morphism#AsElementsOfGaloisGroup}{here}) as elements of [[Galois group]]

\begin{displaymath}
L_{K,\sigma} \colon s
  \mapsto
  \underset{\mathfrak{p}}{\prod}
  det
  \left(   
    id - (N (\mathfrak{p}))^{-s}
    \sigma(Frob_{\mathfrak{p}})
  \right)^{-1}
  \,
\end{displaymath}
(\hyperlink{Gelbhart84}{e.g. Gelbhart 84, II.C.2}, \hyperlink{Snyder02}{Snyder 02, def. 2.1.3}).

\begin{quote}%
discussion of ramified primes needs to be added


\end{quote}
For $\sigma = 1$ the [[trivial representation]] then the Artin L-function reduces to the [[Dedekind zeta function]] (see \hyperlink{RelationToDedekindZeta}{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 \emph{[[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 \emph{[[Artin reciprocity]]}.

For $\sigma$ any $n$-dimensional representation for $n \geq 1$ then the [[conjecture]] of \emph{[[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 \hyperlink{Gelbhart84}{Gelbhart 84, pages 5-6}).

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{for_irreducible_representations__artins_conjecture}{}\subsubsection*{{For irreducible representations -- Artin's conjecture}}\label{for_irreducible_representations__artins_conjecture}

\emph{Artin's conjecture} is the statement that for a nontrivial \emph{[[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. (\hyperlink{RamMurty94}{Ram Murty 94, p. 3})

\begin{quote}%
or rather with at most a pole at $s = 1$ \hyperlink{MurtyMurty12}{Murty-Murty 12, page 29 in chapter 2}


\end{quote}
\hypertarget{ForInducedRepresentations}{}\subsubsection*{{For induced representations}}\label{ForInducedRepresentations}

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:

\begin{displaymath}
L_{K,{Ind_H^{G}\sigma}} = L_{L^H, \sigma}
  \,.
\end{displaymath}
(e.g. (\hyperlink{MurtyMurty12}{Murty-Murty 12, equation (2) in chapter 2})).

\hypertarget{RelationToDedekindZeta}{}\subsubsection*{{Relation to the Dedekind zeta function}}\label{RelationToDedekindZeta}

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]] $\zeta_K$:

\begin{displaymath}
L_{L,1} = \zeta_L
  \,.
\end{displaymath}
If $L/K$ is a [[Galois extension]], the by the behaviour of Artin L-functions for induced representation as \hyperlink{ForInducedRepresentations}{above} this is also the Artin L-function of $K$ itself for the [[regular representation]] of $Gal(L/K)$

\begin{displaymath}
\zeta_L
  =
  L_{L,1}
  = 
  L_{K,{Ind_1^{Gal(L/K)}1}}
\end{displaymath}
(e.g. (\hyperlink{MurtyMurty12}{Murty-Murty 12, below (2) in chapter 2}))

\hypertarget{AnalogyWithSelbergZeta}{}\subsubsection*{{Analogy with Selberg/Ruelle zeta-functions}}\label{AnalogyWithSelbergZeta}

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 \emph{[[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 (\hyperlink{Brown09}{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 \emph{[[Selberg zeta function]]} and \emph{[[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 \emph{\href{Selberg+zeta+function#AnalogyWithArtinLFunction}{Selberg zeta function -- Analogy with Artin L-function}} and at \emph{\href{Ruelle+zeta+function#AnalogyToTheArtinLFunction}{Ruelle zeta function -- Analogy with Artin L-function}} for more on this.

See also (\hyperlink{Brown09}{Brown 09, page 6}, \hyperlink{Morishita12}{Morishita 12, remark 12.7}).

(The definition also has some similarity to that of the [[Alexander polynomial]], see at \emph{[[arithmetic topology]]}.)

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

The original article is

\begin{itemize}%
\item [[Emil Artin]], \emph{\"U{}ber eine neue Art von L Reihen}. Hamb. Math. Abh. 3. (1923) Reprinted in his collected works, ISBN 0-387-90686-X. English translation in (\hyperlink{Snyder02}{Snyder 02, section A})

\end{itemize}
Reviews include

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Artin_L-function}{Artin L-function}}


\item M. Ram Murty, V. Kumar Murty, \emph{Non-vanishing of L-functions and applications}, Modern Birkh\"a{}user classics 2012 (\href{http://www.beck-shop.de/fachbuch/leseprobe/9783034802734_Excerpt_001.pdf}{chapter 2 pdf})


\item [[Noah Snyder]], \emph{Artin L-Functions: A Historical Approach}, 2002 (\href{http://www.math.columbia.edu/~nsnyder/thesismain.pdf}{pdf})



\end{itemize}
and in the context of the [[Langlands program]]

\begin{itemize}%
\item [[Stephen Gelbart]], \emph{An elementary introduction to the Langlands program}, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177--219 (\href{http://www.ams.org/journals/bull/1984-10-02/S0273-0979-1984-15237-6/}{web})

\end{itemize}
Further development includes

\begin{itemize}%
\item M. Ram Murty, \emph{Selberg's conjectures and Artin -functions}, Bull. Amer. Math. Soc. 31 (1994), 1-14 (\href{http://www.ams.org/journals/bull/1994-31-01/S0273-0979-1994-00479-3/home.html}{web})

\end{itemize}
The analogy with the [[Selberg zeta function]] is discussed in

\begin{itemize}%
\item Darin Brown, \emph{Lifting properties of prime geodesics}, Rocky Mountain J. Math. Volume 39, Number 2 (2009), 437-454 (\href{http://projecteuclid.org/euclid.rmjm/1239113439}{euclid})


\item [[Masanori Morishita]], section 12.1 of \emph{Knots and Primes: An Introduction to Arithmetic Topology}, 2012 (\href{https://books.google.co.uk/books?id=DOnkGOTnI78C&pg=PA156#v=onepage&q&f=false}{web})



\end{itemize}
The analogies between Alexander polynomial and L-functions and touched upon in

\begin{itemize}%
\item Ken-ichi Sugiyama, \emph{The properties of an L-function from a geometric point of view}, 2007 \href{http://geoquant2007.mi.ras.ru/sugiyama.pdf}{pdf}; \emph{A topological $\mathrm{L}$ -function for a threefold}, 2004 \href{http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1376-12.pdf}{pdf}

\end{itemize}
[[!redirects Artin L-functions]]



\end{document}