#10973: Integral points on elliptic curves over number fields
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Reporter: justin | Owner: cremona
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-6.1
Component: elliptic curves | Resolution:
Keywords: sd32 | Merged in:
Authors: Justin Walker, | Reviewers:
Aly Deines, Jennifer Balakrishnan | Work issues:
Report Upstream: N/A | Commit:
Branch: | 7f5036284d4b31fb81bd69a9bf0a1927c70f3206
u/cremona/trac10973intpts | Stopgaps:
Dependencies: |
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Comment (by cremona):
Replying to [comment:50 chapoton]:
> Why is this "needs works" ?
Because I am actively working on it! Seriously, comparing in detail
Smart's paper and book and Nook's account in his thesis, I have found two
things: (1) Smart does not deal properly with torsion generators, Nook
does and that is in this code but I think I might be able to do better;
(2) more seriously, Smart derived an upper bound for {{{|x(P)^2\psi(P)|}}}
(relative to each embedding) which only valid for points with {{{|x(P)|}}}
bounded below by some quantity (in the appropriate embedding), and this
means that there are points, possible integral, which fail this condition
and need to be considered separately. Over Q that is easily done by
searching on a finite x-interval, but over a general number field it is
not clear to me that the "missing" points are in a finite region.
However, I am working on a bound (which certainly exists) for
{{{|x(P)^2\psi(P)|}}} valid for all complex points with any embedding, and
when that is done I will be changing some of the constants in the code.
I am not expecting the output to be different in any of the examples (but
who knows), but I cannot let this in when I have identified but not yet
fixed this mathematical bug!
--
Ticket URL: <http://trac.sagemath.org/ticket/10973#comment:51>
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