Artin L-function

In mathematics, Artin L-functions are meromorphic functions associated to Galois extensions of global fields. More precisely, if K/k is such an extension with Galois group G, then to each representation of G on a finite dimensional complex vector space, there is an associated Artin L-function. When K and k are algebraic number fields, Artin L-functions generalize Dedekind zeta functions, which are just the Artin L-functions associated to trivial extensions. Artin L-functions are important because they encode arithmetic information about the extension. For instance, the Stark conjectures predict that the coefficient of the leading term of the Taylor series expansion of an Artin L-function around s=0 provides information about the units of the field K. This can be viewed as a generalization of the analytic class number formula.

Definition

Let K/k be Galois extension of global fields, and let $\rho$ be a representation of the Galois group $\scriptstyle G=\mathrm {Gal} (K/k)$ on a finite dimensional complex vector space V. The Artin L-function associated to $\rho$ is defined by the Eulerian product

$L(K/k,\rho ,s)=\prod _{\mathfrak {p}}{\frac {1}{\det \left(1-\varphi _{\mathfrak {P}}{\mathfrak {N}}\left({\mathfrak {p}}\right)^{-s};V^{I_{\mathfrak {P}}}\right)}}.$

The product extends over the set of prime ideals of k, and ${\mathfrak {P}}$ is an arbitrarily chosen prime ideal of K dividing ${\mathfrak {p}}$. Also, $\varphi _{\mathfrak {P}}$ is the Frobenius automorphism in G associated to ${\mathfrak {P}}$, and $I_{\mathfrak {P}}$ is the corresponding inertial group. The determinant in the definition is independent of the choice of the prime ideal ${\mathfrak {P}}$. Also, although the Frobenius automorphism is only determined up to an element of $I_{\mathfrak {P}}$, the determinant is independent of this choice.