Re: L-functions twisted by character

[apacetti@dm.uba.ar]

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.

Ariel


> 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.
>