Bill Allombert on Wed, 21 Oct 2015 23:41:08 +0200 |
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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. ? P=quadhilbert(-47);Q=nfsplitting(P); ? 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 ? P=quadhilbert(-47);Q=nfsplitting(P); ? 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