#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 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.
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?
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/13951#comment:14>
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