Re: L-functions twisted by character

[Karim Belabas <Karim.Belabas@math.u-bordeaux.fr>]

* apacetti@dm.uba.ar [2017-12-11 16:23]:
> What you should do is to first compute the twist and its minimal model
> (this recovers the missing local factors), and then compute the L-series
> value.
> 
> If you want to avoid doing this, you can recover the missing local factor
> (if any) by  counting points over field extensions. This is explained in
> an article by Tim and Vlad Dokchitser.

Sure: that's precisely what Bill suggested to do when we are dealing with the
L-function of an elliptic curve.

But his implementation also allows to twist *arbitrary* (motivic)
L-functions by a Dirichlet character (of arbitrary order), as long as
the conductors are coprime. And this is useful as well, even with the
coprimality restriction :-)

Cheers,

    K.B.

> > On Sat, Sep 17, 2016 at 10:16:02AM +0200, Bill Allombert wrote:
> >> On Fri, Sep 16, 2016 at 11:14:09AM +0200, Jeroen Demeyer wrote:
> >> > Does PARI support L-functions (in particular of elliptic curves)
> >> twisted by
> >> > a character? I couldn't find it explicitly in the manual, but I might
> >> have
> >> > missed something.
> >>
> >> No it is not implemented. The problem is to compute correctly the
> >> conductor and the Euler factors at the bad primes.
> >
> > I added a function lfuntwist which allows to twist by an arbitrary
> > primitive Dirichlet character, however it requires the conductor of
> > the character to be prime to the conductor of the L-function, which
> > avoid the above issue.
> >
> > In this example, we twist the L function of an elliptic curve by a
> > character of order 3 and of conductor 7.
> >
> > ? E=ellinit("11a1");
> > ? nf=nfinit(polsubcyclo(7,3,'a));
> > ? E2=ellinit(E[1..5],nf);
> > ? L=lfuntwist(E,Mod(2,7));
> > ? lfun(E,2)*norm(lfun(L,2))-lfun(E2,2)
> > %24 = -7.560250692031083222E-34
> >
> > Due to the restriction, for L function of elliptic curves twisted by a
> > quadratic character it is better to twist the curve instead.
> >
> > Cheers,
> > Bill.
--
Karim Belabas, IMB (UMR 5251)  Tel: (+33) (0)5 40 00 26 17
Universite de Bordeaux         Fax: (+33) (0)5 40 00 21 23
351, cours de la Liberation    http://www.math.u-bordeaux.fr/~kbelabas/
F-33405 Talence (France)       http://pari.math.u-bordeaux.fr/  [PARI/GP]
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