#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 robertwb):
Replying to [comment:2 cremona]:
> I have had a quick look and will go through this in more detail later
(after #8828 is completed, probably). I spent a long time on my C++
implementation of this (over QQ but the algorithm is general) so am quite
familiar with the details.
>
> Here are two references you should give: [1] S. Siksek "Infinite
descent on elliptic curves", Rocky Mountain J of M, Vol 25 No. 4 (1995),
1501-1538. [2] M. Prickett, "Saturation of Mordell-Weil groups of
elliptic curves over number fields", U of Nottingham PhD thesis (2004),
http://etheses.nottingham.ac.uk/52/.
Ah, those look like good references to read too :).
> Martin Prickett implemented this in Magma, but the code was very slow
and hard to read so it never got incorporated into Magma releases.
>
> Incidentally, it was for this that I implemented group structure for
curves over GF(q) in the first place! In my C++ implementation I cache a
lot of the information of this group structure so that when you do
p-saturation for larger and larger p, the structures are already there.
The way I do it is consider many p at once, and for each curve over GF(q)
I see which primes in my set it could help with, though this won't scale
as far. I'm sure there's still lots of room for improvement.
> A good example is to take one of those curves of very high rank: I
think I once successfully p-saturated the rank 24 curve at all p <
{{{10^6}}} (the bound was totally out of reach, something like
{{{10^100}}}).
That reminds me--I was wondering if there's any way to go from min(h(P))
to a bound on the regulator for rank > 1.
> Another point which might be useful over number fields: it suffices to
use degree one primes to reduce modulo.
Good point. Those get pretty rare for large degree number fields though,
right?
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8829#comment:3>
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