#8829: Saturation for curves over number fields.
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Reporter: robertwb | Owner: cremona
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-4.4.1
Component: elliptic curves | Keywords:
Author: | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Comment(by cremona):
You might also like to look at my C++ code which is in eclib, in
src/qcurves. I can point to the right files if it is not clear. In case
you wonder, "TLSS" stands for "Tate-Lichtenbaum-Samir_Siksek" since I use
the TL map when the p-torsion in E(GF(q)) is not cyclic and Samir's
original method when it is. Samir only used reduction modulo primes where
p exactly divided the order, and in particular for which the reduction had
cyclic p-part. But Martin and I discovered that this can fail when there
is a p-isogeny. Here, fail means in the sense that there can exist points
which are not multiples of p in E(QQ) but which map to zero in E(GF(q))/p
for all q.
In MP's thesis he proves that this cannot happen if you use all q, or all
but a finite number, or all but a finite number of degree 1 primes, ....
some of these results we then found had been proved elsewhere (3 or 4
times, independently, within 3 or 4 years!). But it can happen if you
leave out the q for which the quotient has non-cyclic p-part.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8829#comment:4>
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