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/ -~----------~----~----~----~------~----~------~--~---
