Re: Making idealmoddivisor_aux public

[Robert Harron <robert.harron@gmail.com>]

Thank you for your insight. My goal had indeed been to implement these 
computations using the representation of pairs [I, t] you describe (and 
as suggested in Henri Cohen's book), but I hadn't figured out how to do 
that with the pari functions I could find. I'll take some time now to 
figure that out.

Best,

Rob

On 8/16/16 1:09 AM, Karim Belabas wrote:
> * Robert Harron [2016-08-16 12:12]:
> > Indeed, the implementation I've written so far in Sage uses the discrete
> > logs of the elements for computations, and it is true that in terms of
> > working with Hecke characters this, together with bnrisprincipal suffices.
> > But the implementation I've written mimics the current implementation for
> > the class group which keeps track of not only the discrete logs, but also
> > representative ideals. And like in the class group code, upon multiplication
> > an equivalent 'small' ideal representative is found.
>
> Sure, you don't have coprimality or sign conditions there: you can get
> away with "actually small" representatives (controlled by the field
> discriminant). Now try Algorithm 4.3.4 with "nice" totally real fields
> of degree 20 or so with [K:Q] sign conditions and a non-trivial
> modulus...
>
> > So, while one can get away without reducing and still compute with
> > Hecke characters, I still want to be able to provide the user with
> > representatives of the ray classes.
>
> What for ? My point is that those are going to be relatively costly,
> and much larger than they need to be. I honestly have never seen any
> actual use for them, and I've been implementing CFT-related algorithms
> for some time now.
>
> In practice, it is better to have
>
> 1) exact representations (in an efficient format such as my formal
> products, aka compact representations) than trying to "reduce" modulo
> complicated rules. Then you can map locally to wherever group you realy
> need them at 0 cost, e.g. R^*/(R^*)^2 for signs, or
> (Z_K/\mathfrak{p}^k)^* for CRT-based representation of (Z_K/f)^*.
>
> 2) local computations (think Chinese Remainder Theorem) rather than
> global representatives. (As a bonus, it is technically simpler, and
> you don't have to handle global obstructions.)
>
> > We're you saying that idealmul and idealred themselves allow for finding
> > small representatives of an ray class? I couldn't understand how to do that.
>
> Not quite. To compute exactly with ideals in a way that can be
> efficiently mapped to ray class groups, or other situations.
>
> > Or wait, I see now that you mention a public function idealmoddivisor, but
> > it looks like that function has disappeared from the documentation, maybe
> > even from being publicly available.
>
> It's still there, both in documentation and headers. But useless, as I
> tried to explain :-)
>
> > I found it in a google search in pari 2.5.
>
> Try extended help:
>
> (12:54) gp > ???idealmoddivisor@        \\ @ means: search in the whole manual
>
> Chapter 6:
> ==========
>   Ideal reduction, high level
>
> (That's 2.8.0, or 'master' GIT branch.)
>
> Cheers,
>
>     K.B.
> --
> Karim Belabas, IMB (UMR 5251)  Tel: (+33) (0)5 40 00 26 17
> Universite de Bordeaux         Fax: (+33) (0)5 40 00 69 50
> 351, cours de la Liberation    http://www.math.u-bordeaux.fr/~kbelabas/
> F-33405 Talence (France)       http://pari.math.u-bordeaux.fr/  [PARI/GP]
> `