Re: Making idealmoddivisor_aux public

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

Thank you for your replies.

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. 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.

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. 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. I found it 
in a google search in pari 2.5.

Thank you,

Rob

On 8/15/16 11:26 PM, Karim Belabas wrote:
> * Karim Belabas [2016-08-16 10:59]:
> > * Robert Harron [2016-08-16 05:30]:
> > > I'm writing code for computing ray class groups and Hecke characters in Sage
> > > mostly by wrapping the functions in the pari c library. One particular
> > > function I've had to write myself is the one that takes an ideal I and
> > > returns a 'small' ideal equivalent I modulo the given modulus (Algorithm
> > > 4.3.4 of Henri Cohen's Advanced topics book). I did however find that there
> > > is a function in the pari source code that does this: idealmoddivisor_aux in
> > > the file buch3.c. I was wondering if it would be possible to make a public
> > > version of this function in the pari library, perhaps just adding a modulus
> > > argument to idealred. It's pretty essential for making efficient
> > > computations in the ray class group and I'm sure the pari library version
> > > would be quicker than what I've written!
> >
> > This function is certainly not essential; in fact, it was implemented as
> > a proof of concept then discarded for uselessness: it's a red herring.
> > The documentation of idealmoddivisor(), the public function doing what
> > you require, comments as follows: "a proof-of-concept implementation,
> > useless in practice".
> [...]
> > I agree that all this should be better documented :-)
>
> Of course, in order to "compute in ray class groups", the simplest (and
> usually best) solution is to directly use bnrisprincipal() and express
> everything in terms of discrete logs. That's how Hecke characters
> (currently of finite order) are implemented in PARI.
>
> In short, we'll need to know more about what you actually need, and why.
>
> 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]
> `