John Cremona wrote: > I agree with Nils! > +1
Jaap > John > > On 9/6/07, Nils Bruin <[EMAIL PROTECTED]> wrote: >> I would be very disappointed if this approach would be taken. Sure, it >> has to be straightforward and convenient to choose the level of rigour >> desired in class group computations, but by default results should be >> unconditional. It would be a real shame if SAGE, which tries to up the >> level of trustworthiness of computational proofs by having the source >> open for inspection would at the same time lower the level by tacitly >> assuming conjectures. >> >> Would you advocate that ranks of elliptic curves should be computed >> analytically because BSD is a well-established conjecture without >> known counter examples? Should we be computing ranks of arbitrary >> abelian varieties analytically because BSD doubtlessly holds for these >> too? >> >> It makes a big difference whether you prove "subject to GRH, this >> curve does not have rational points" or "this curve does not have >> rational points". >> >> By allowing unproved default behaviour in Sage for this, you will set >> a very bad precedent and you risk getting a bad word-of-mouth >> reputation of the kind "Oh you used SAGE for that? Better verify those >> results. They assume conjectures". Pari definitely suffered from that >> effect due to their "bachbound/40" and made me avoid it. >> >> I also don't agree that class group computations are virtually >> infeasible without conjectures. If the minkowskibound (or if someone >> wants to work out an improvement, the improved bound) is around, say >> 10^6, one can easily check the factorisation of all ideals up to that >> norm, even for degree 12 fields. This actually does happen for certain >> applications. >> >> On Sep 5, 10:00 pm, "William Stein" <[EMAIL PROTECTED]> wrote: >>> On 9/5/07, Bill Hart <[EMAIL PROTECTED]> wrote: >>> >>>> Are you asking me? I'm not even sure I understand all the different >>>> options Magma offers yet!! >>>> Basically I don't see any problem with doing the computation the way >>>> Pari does it, whatever that equates to in Magma. >>> I'm specifically asking you, but would like to get feedback from >>> others. It seems to me that with most class number calculations, >>> one either assumes conjectural bounds or they all become >>> virtually impractical. >>> >>> Unless anybody seriously objects, I'm going to set the default >>> to False in SAGE, based on your comments below and my own >>> personal inclinations. >>> >>> >>> >>>> *IF* I understand it correctly, from Magma's point of view, Pari uses >>>> the bound BachBound/20 or BachBound/40 (depending on whether the field >>>> is quadratic or not) and uses the "subgroup" level of proof. >>>> If you do demand "proof = true" then I believe you can do better than >>>> Minkowski's bound. Zimmert's bounds are also unconditional and better >>>> than Minkowski. >>>> My reason for feeling that Pari's approach is OK is that in many cases >>>> you know that hR is at least correct from the way the algorithm works, >>>> if not it has checked up to a fairly conservative bound. You really >>>> can't go too wrong (though obviously it is not excluded >>>> theoretically). Furthermore there is a conjecture which gives an even >>>> lower bound than the bound used by Pari. There are also no known >>>> exceptions. >>>> The thing that surprises me the most about this subject is that it >>>> probably isn't possible to compute the class number faster than the >>>> class group structure, especially in the generic case. One could get >>>> hR from the analytic class number formula, but to compute R one would >>>> need fundamental units, but computing those is essentially equivalent >>>> to what we are doing. Anything else I think of seems to have the same >>>> complexity except in very special cases where one knows something else >>>> in advance. >> > > --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://sage.scipy.org/sage/ and http://modular.math.washington.edu/sage/ -~----------~----~----~----~------~----~------~--~---
