#13951: (non)archimedian_local_height broken for rational points on elliptic
curves
over Q
-----------------------------------+----------------------------------------
Reporter: pbruin | Owner: cremona
Type: defect | Status: needs_review
Priority: major | Milestone: sage-5.10
Component: elliptic curves | Resolution:
Keywords: local heights | Work issues:
Report Upstream: N/A | Reviewers:
Authors: Peter Bruin | Merged in:
Dependencies: #12509, #13953 | Stopgaps:
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Comment (by cremona):
Replying to [comment:10 pbruin]:
> Replying to [comment:8 cremona]:
> > I am looking at this now. I note that the two prerequisites are
already merged in 5.10.beta3/4 and plan to test on 5.10.rc0.
> >
> > One preliminary comment: the little function to compute the local
degree at an archimedean place by testing if the image of the generator
under the embedding has 0 imaginary part looks very ugly to me (and it is
possible that I wrote it). Surely there is a better way? I looked to see
what I did for the period functions in period_lattice.py to test whether
an embedding was real or not: there, the given embedding is first refined
to an embedding into AA if real or QQbar if not, using
sage.rings.number_fields.refine_embedding, which in turn tests whether the
codomain of the embedding passes sage.rings.real_mpfr.is_RealField().
Would that be better?
> >
> >
> One easy solution would be to rely on the fact that K.places() returns
the r_1 real embeddings followed by the r_2 complex embeddings:
>
>
> {{{
> r1, r2 = K.signature()
> pl = K.places()
> return (sum(self.archimedian_local_height(pl[i], prec) for i in
range(r1))
> + 2 * sum(self.archimedian_local_height(pl[i], prec) for i in
range(r1, r1 + r2))) / K.degree()
>
> }}}
> This leaves the job of distinguishing real and complex places to the
NumberField code. I just tested this and it appears to work
I like that a lot. Since I am already working on a second patch, I will
put that in. Thanks.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/13951#comment:11>
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