L-functions/ArtinLFunctions - L-Functions Wiki

Contents

Report of the Group
Initial Definitions
Euler product and functional equation
Properties of the Artin L-function
References
Authorship

Report of the Group

Artin L-functions are important because of their natural structure, relationships with many other L-funtions, such as Dirichlet L-functions, Hecke L-functions and the Dedekind zeta function, and because of the Langlands programme.

There have been very few reports of computations of Artin L-functions. The known ones are reported below. There are very few published tables. Mostly research has been into special types with small degree polynomials (less say than five) and small conductors. To work these out, tables of fields and tables of groups and group characters have often been consulted. Much of the existing work has been an investigation of Artin's holomorphy conjecture.

A start is being made with the Mathematica package ZetaPack (http://www.math.waikato.ac.nz/~kab) to design generic programs to compute Dirichlet coefficients and values for these L-functions for number fields over the rationals. Initial algorithms are expected to be slow, except when field parameters are small.

Initial Definitions

Here we give some definitions needed to define the L-function and functional equation, then give some properties.

1. If L/K is a Galois extension with group\epsilon G and P\subset \mathcal{O}_L a prime ideal then the decomposition group of P is the subgroup

D_P = \{ g\in G : P^g = P \}.

2. If L/K is a Galois extension with group G and P\subset \mathcal{O}_L a prime ideal then the inertia group of P is the subgroup

I_P = \{ g \in G : x^g \equiv x\bmod P ~\text{for all $x\in \mathcal{O}_K$}\}.

3. The Frobenius automorphism of L/K attached to a prime ideal p\subset \mathcal{O}_K is a member {\rm Fr}_P of the decomposition group D_P which satisfies

x^{{\rm Fr}_P}\equiv x^{N_K(p)} \bmod P

for all x\in \mathcal{O}_K.

4. The p-adic inertia group or ramification group of order i with respect to the prime p\subset K is the subgroup G_i\subset Gal(L/K) which acts trivially on O/P^{i+1}, where P is a prime ideal sitting above p.

5. The Artin character a is defined by

a(\tau)= -i ~\text{if $\tau\in G_{i-1}-G_i$ where $\tau\neq 1$}

where G_i is the i^{\text{th}} ramification group, and a(1) is defined so that

\sum_{\tau\in G_0} a(\tau)=0.

6. The local Artin conductor for a prime ideal p\subset \mathcal{O}_K attached to a character \chi is

f_p(\chi,L/K)= \sum_{i=0}^\infty \frac{g_i}{g_0}{\rm codim} V^{G_i}

where {\rm codim} V^{G_i}= \dim V - \dim V^{G_i}, G_i is the i^{\text{th}} ramification group, and g_i=|G_i| is its order.

7. The global Artin conductor or simply Artin conductor is the ideal

f(p,L/K) := \prod_p p^{f_p(\chi,L/K)}

where the product is over all prime ideals p\subset \mathcal{O}_K.

8. If V is a representation of G and H\subset G a normal subgroup, then

V^H = \{ v\in V : h.v=v \text{for all $h\in H$}\}

is a representation of G/H called the quotient representation.

9. The Artin root number of is a complex number \epsilon(\chi,L/K)=W(\chi), of modulus 1, which balances the functional equation for the completed Artin L-function \Lambda(s,\rho,L/K). Then the functional equation reads \Lambda(1-s)=\epsilon(\chi)\Lambda(s) so

\Lambda(s)=\Lambda(1-(1-s))=\epsilon(\chi)\Lambda(1-s)=\epsilon(\chi)^2\Lambda(s)

leading to the surprising conclusion that W(\chi)^2=1, and thus \epsilon(\chi)=\pm 1. The breaking down on \epsilon(\chi) into a product of local roots numbers \epsilon_p(\chi) is beyond the scope of these definitions.

10. An infinite prime of an extension K of \mathbb{Q} is a valuation v:K\rightarrow [0,\infty) which is a composite of either a real embedding of K\rightarrow \mathbb{R} or a pair of complex conjugate embeddings K\rightarrow \mathbb{C} with the normal complex modulus. These valuations are Archimedian.

11. A finite prime of an extension K of \mathbb{Q} is a valuation v:K\rightarrow [0,\infty) is associated with a prime ideal p\subset \mathcal{O}_K via the definition

|x|_p = \frac{1}{N_K(p)^r}

for x\neq 0, where r is the unique integer with x\in p^r and x\not\in p^{r+1}, with |0|_p =0. These valuations are non-Archimedian.

Euler product and functional equation

Let \rho:G\rightarrow GL(V) be a representation of L/K, a Galois extension. Then

Finite primes

Let p be a finite prime in \mathcal{O}_K, ramified or unramified, and \rho a representation with character \chi. Then define

L_p(s,\rho,L/K)=\det (Id- N_K(p)^{-s}(\rho({\rm Fr}_P))|V^{I_P})^{-1}

which is equivalent to

L_p(s,\chi,L/K)=\exp (\sum_{n=1}^\infty \frac{(\chi({\rm Fr}_P))|V^{I_P})}{N_K(p)^{ns}}

where P is a prime ideal in L above p.

Infinite primes

Let v be an infinite prime, i.e. a valuation arising from a real embedding or pair of complex embeddings. Then let

\Gamma_\mathbb{R}(s)= \pi^{-\frac{s}{2}} \Gamma(\frac{s}{2}),

\Gamma_\mathbb{C}(s)= 2(2\pi)^{-s} \Gamma(s)=\Gamma_\mathbb{R}(s)\Gamma_\mathbb{R}(s+1),

and

L_v(s,\chi,K)= \Gamma_\mathbb{R}(s)^{(\dim V^{G_v})}\cdot \Gamma_\mathbb{R}(s+1)^{({\rm codim} V^{G_v})}

if v is real, or

L_v(s,\chi,K)= \Gamma_\mathbb{C}(s)^{\chi(1)}

if v is complex.

Functional equation

Define the completed Artin L-function as

\Lambda(s,V,L/K)= c(V,L/K)^\frac{s}{2}\prod_p L_p(s,V,L/K)

where the product is over all primes, finite and infinite, ramified and unramified, and the so-called exponential factor c(V,L/K) is given by

c(V,L/K)=|{d_K}^{\dim V}_{K}N_{K}({f}(V,L/K))|

Then the functional equation may be written

\Lambda(s,V,L/K) = \epsilon(V,L/K)\Lambda(1-s, V^*,L/K),~\text{or} \Lambda(s,\chi,L/K) = \epsilon(\chi)\Lambda(1-s,\bar{\chi},L/K)

where \epsilon(V,L/K)=\pm 1 is the root number.

Properties of the Artin L-function

1. If K=L=\mathbb{Q} then L(s,\rho,\mathbb{Q}/\mathbb{Q})=\zeta(s) the Riemann zeta function. If \rho is the trivial representation of dimension 1, \chi_0(g)=1 for all g\in G, then L(s,\chi_0, K/\mathbb{Q})=\zeta_K(s), the Dedekind zeta function.

2. L(s,\chi_1+\chi_2, L/K)= L(s, \chi_1,L/K)\cdot L(s,\chi_2,L/K)

3. If K\subset L\subset M, M/K is Galois and L/K is Galois, then L(s,\chi,M/K)=L(s,\chi^*,L/K) where \chi* is the restriction of \chi.

4. If K\subset L\subset M and \chi is a character of Gal(M/L) then the induced character Ind^G_G \chi on Gal(M/K) satisfies L(s,\chi,M/L)=L(s,Ind_G^G\chi,M/K).

References

Booker, A. R. "Artin's conjecture, Turings method, and the Riemann hypothesis", Experiment. Math. 15 (2006), 385-407.

Lagarias, J.C. and Odlyzko, A.M. "On computing the Artin L-functions in the critical strip", Math. Comp. 33 (1979), 1081-1095.

Omar, S. " On Artin L-functions for octic quaternion fields", Experiment. Math. 10 (2001), no. 2, 237--245.

Buzzard,K., Dickinson,M., Shepard-Barron, N. and Taylor, R. "On Icosahedral Artin Representations", Duke. Math. J. 109 (2001), 283-318.

Authorship

This page was written by (primary author) Kevin Broughan, who takes full reponsibility for errors (unless and until they are fixed!).

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