#8828: Lower height bound for elliptic curves
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Reporter: robertwb | Owner: cremona
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-6.2
Component: elliptic curves | Resolution:
Keywords: | Merged in:
Authors: Robert Bradshaw, | Reviewers:
John Cremona | Work issues:
Report Upstream: N/A | Commit:
Branch: | ccd138c3e9aeab0c35ccc6e5a5d3011fe21b47b4
u/cremona/ticket/8828 | Stopgaps:
Dependencies: |
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Comment (by cremona):
I have finished workin on height.py, adding a large amount of
documentation and examples. There were some bugs revealed by this, which
have been fixed. One was in the function {{{non_neg_region}}} and was a
simple slip, I forget the details. The second was due to wrong
normalisation of period lattice basis (for real embeddings): L.basis()
gives w1, w2 with w1 real, while L.normalised_basis() gives w1,w2 with w2
minimal and tau=w1/w2 in the fundamental region. To avoid this catching
people out in future I added a method tau() to the period lattice class.
The effect here is that in some of the tests the error is now better than
it was; but also the test results (for fk and wp) are different since
they work with respect to the normalised lattice [1,tau] and so are
different when the correct value of tau is used!
I added more examples from the papers cited, and also give more specific
references to those papers throughout, especially the paper [TT] by Nook
(Thotsaphon Thonjungthug); the theory here was developed by himself with
me and Samir Siksek while Nook was our PhD student.
It seems to be four years since Robert Bradshaw implemented this (Nook
himself implemented it too, in Magma) so it seem rather unreasonable to
ask him to look at it again. Peter Bruin would be a good person, since
his own recent paper improves Nook's results (and makes some of this
redundant perhaps?)
I will set this to "needs review" as soon as I have added a couple of
missing doctests in the file {{{period_lattice_region.pyx}}} to bring the
two files up to 100%.
--
Ticket URL: <http://trac.sagemath.org/ticket/8828#comment:29>
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