#13951: (non)archimedian_local_height broken for rational points on elliptic
curves
over Q
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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 pbruin):
Replying to [comment:14 cremona]:
> Replying to [comment:13 pbruin]:
> > And maybe the global height function should also have a parameter
"weighted", where True means no division by the degree of the number
field. Instead of "weighted" the parameter could be called something like
"absolute" or "normalised", where True means that we ''do'' divide by the
degree.
>
> OK, I am working on it. For non-arch. places the default, only relevant
for a single place, is weighted=False but when it adds up all the non-
arch contributions it puts in the weightings as that is needed for the
global height. When weighting is False the local height is divided by the
local degree. Now for arch. places there is currently no weighting for
individual places but we multiply by the local degree when adding up all
the arch. contributions. So I think that the consistent thing to do would
be have weighting=True mean that at complex places the local height would
be multiplied by 2, then again the total arch. contribution would be
obtained by adding up the individual ones, weighted, but the default for a
single place would be to have weighting=False and return the same as now.
It seems to me that the parameter "weighted" still makes sense if v =
None: we should take the weighted sum of the local heights at all
(non-)Archimedean places, and if weighted = False, we divide this sum by
the degree of K.
Although it would certainly be good to have a parameter "weighted" for
archimedean_local_height, the drawback is that we would again need to
distinguish between real and complex places. With the above way of
eliminating the local_degree function, this is only avoided because we
know which one it is because of our position in the list of all places.
> The only thing about all that which puzzles me is that in the non-arch.
case we divide by the local degree when weighting=False, while for arch.
places we multiply by it when weighting=True. This suggests to me that
our conventions for the non-weighted local heights are inconsistent. What
do you think?
The current conventions for the return values of
(non)archimedian_local_height are consistent. The difference is that the
calculation is done differently: for v Archimedean we compute with the
smaller of the two all the time and multiply by the local degree at the
end if necessary, while for v non-Archimedean we compute with the larger
of the two and divide by the local degree at the end if necessary.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/13951#comment:15>
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