Bill Allombert on Wed, 21 Oct 2015 23:41:08 +0200

 GP interface for computing Artin L functions

```Dear PARI developers,

We just added to master a new function lfunartin() to compute
Artin L functions.
This is based on a GP script by Charlotte Euvrard.
Currently, the representation needs to be given explicitly

This is the documentation:

lfunartin(nf,gal,M,n):

Returns the Ldata structure associated to the Artin L-function associated to the
representation  rho of the Galois group of the extension K/Q,  defined over the cyclotomic field
Q(zeta_n),   where  nf is the nfinit structure associated to K,  gal is the galoisinit structure
associated to K/Q, and M is the vector of the image of the generators G.gen by rho. The elements
of  M  are matrices with polynomial entries,  whose variable is understood as the complex number
exp(2 i Pi/n).

In  the  following  example  we build the Artin L-functions associated to the two irreducible
degree-2 representations of the dihedral group D_{10} defined over Q(zeta_5),  for the extension
H/Q  where  H  is the Hilbert class field of Q(sqrt{-47}).   We show numerically some identities
involving Dedekind zeta functions and Hecke L series.

? N=nfinit(Q);G=galoisinit(N);
? L1=lfunartin(N,G,[[a,0;0,a^-1],[0,1;1,0]],5);
? L2=lfunartin(N,G,[[a^2,0;0,a^-2],[0,1;1,0]],5);
? lfun(1,1)*lfun(-47,1)*lfun(L1,1)^2*lfun(L2,1)^2 - lfun(Q,1)
%5 ~ 0
? lfun(1,1)*lfun(L1,1)*lfun(L2,1) - lfun(P,1)
%6 ~ 0
? bnf=bnfinit(x^2+47);bnr=bnrinit(bnf,1,1);
? lfun([bnr,[1]],1) - lfun(L1,1)
%7 ~ 0
? N=nfinit(Q);G=galoisinit(N);
? L1=lfunartin(N,G,[[a,0;0,a^-1],[0,1;1,0]],5);
? L2=lfunartin(N,G,[[a^2,0;0,a^-2],[0,1;1,0]],5);
? lfun(1,1)*lfun(-47,1)*lfun(L1,1)^2*lfun(L2,1)^2 - lfun(Q,1)
%5 ~ 0
? lfun(1,1)*lfun(L1,1)*lfun(L2,1) - lfun(P,1)
%6 ~ 0
? bnf=bnfinit(x^2+47);bnr=bnrinit(bnf,1,1);
? lfun([bnr,[1]],1) - lfun(L1,1)
%7 ~ 0

? lfun([bnr,[2]],1) - lfun(L2,1)
%8 ~ 0
? lfun(1,1)*lfun([bnr,[1]],1)*lfun([bnr,[2]],1) - lfun(P,1)
%9 ~ 0

The  first identity is the factorisation of the regular representation of D_{10},  the second
the  factorisation  of  the  natural  representation of D_{10}\subset S_5,  the next two are the
expressions of the degree-2 representations as induced from degree-1 representations.   The last
one is a collorary of the others.

The library syntax is GEN lfunartin(GEN nf, GEN gal, GEN M, GEN n).

Cheers,
Bill

```