#15115: correct set of points at infinity for hyperelliptic curve
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Reporter: mstreng | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-5.12
Component: geometry | Resolution:
Keywords: hyperelliptic | Merged in:
curve points at infinity | Reviewers:
Authors: Marco Streng | Work issues: doctest the rest of
Report Upstream: N/A | the sage library, check how
Branch: | documentation looks, add example
Dependencies: #15108 | from ticket description to patch
| Commit:
| Stopgaps:
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Comment (by nbruin):
Replying to [comment:4 mstreng]:
> One more pessimistic remark though: elliptic curves over the p-adics
inherit from hyperelliptic curves, so switching to weighed projective
space with weights (1, g+1, 1) for all hyperelliptic curves will lead to
trouble.
Yes, if we're going to give our curves by projective models then elliptic
curves should probably be given by a cubic model in ordinary projective
space. That means that the common stuff between p-adic hyperelliptic
curves and p-adic elliptic curves needs to be factored out into an
object/interface that does not care about projective models.
> Of course the fake points at infinity in this patch are an ugly hack,
No, they are just wrong!
> but using a singular model gives mathematically incorrect answers (as in
#11980),
But in those cases the answers can be made correct by rephrasing them in
terms of "birational" rather than "isomorphic" and if necessary in terms
of "places of degree 1" rather than points.
> and implementing the whole weighted projective spaces thing takes time.
So which is the lesser of three evils?
reporting points that don't lie on the projective model advertised is
definitely the worst of the three in my view. Having a model where there's
no 1-1 correspondence between the degree 1 places and the (possibly
singular) rational points is perhaps inconvenient, but not intrinsically
wrong. Eventually having both agree is of course what we should strive
for.
--
Ticket URL: <http://trac.sagemath.org/ticket/15115#comment:6>
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