Robert Harron on Tue, 16 Aug 2016 21:56:39 +0200 |
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Re: Making idealmoddivisor_aux public |
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] `