Xavier Roblot on Thu, 22 Oct 2015 18:03:59 +0200 |
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Re: GP interface for computing Artin L functions |
> On 22 Oct 2015, at 14:50, Bill Allombert <Bill.Allombert@math.u-bordeaux.fr> wrote: > > On Thu, Oct 22, 2015 at 02:18:41PM +0200, Xavier Roblot wrote: >> Hi Bill, >> >>> 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). >> >> This is great news and that will be very useful! Do you think it could be >> possible to also specify a finite set of prime ideals of K (maybe by >> providing an integral ideal of K) that should be excluded from the Euler >> product defining the L-function? This kind of L-functions are very often used >> in the context of Stark conjectures. > > Does such L functions satisfy a functional equation compatible with the lfun > interface ? Well, I am not sure about that. But, usually, the way to do it is to compute the value of the primitive L-function and then multiply it by the right factor that is relatively easy to compute. In fact, I think I could even probably write the corresponding code if I have your blessing ;) Xavier