Opened 13 years ago
Last modified 5 years ago
#8829 closed enhancement
Saturation for MW-groups of elliptic curves over number fields. — at Version 21
Reported by: | Robert Bradshaw | Owned by: | John Cremona |
---|---|---|---|
Priority: | major | Milestone: | sage-8.1 |
Component: | elliptic curves | Keywords: | saturation |
Cc: | Peter Bruin, Kiran Kedlaya | Merged in: | |
Authors: | Robert Bradshaw, John Cremona | Reviewers: | John Cremona |
Report Upstream: | N/A | Work issues: | |
Branch: | u/cremona/8829 (Commits, GitHub, GitLab) | Commit: | 0e1e35f624edb087d3fb1ddc21226fec7acfafad |
Dependencies: | #8828 | Stopgaps: |
Description (last modified by )
Implementation of saturation of points on elliptic curves over number fields. Originap patch by Robert Bradshaw in 2010, refactored and made into a git branch by John Cremona in 2015.
I also implemented the simple case of E.gens() for E(K) when E/Q and [K:Q] = 2.
Change History (22)
comment:1 Changed 13 years ago by
Status: | new → needs_review |
---|
comment:2 follow-up: 3 Changed 13 years ago by
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/.
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. 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
).
Another point which might be useful over number fields: it suffices to use degree one primes to reduce modulo.
Changed 13 years ago by
Attachment: | 8829-ec-nf-sat.patch added |
---|
comment:3 Changed 13 years ago by
Replying to 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 like10^100
).
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?
comment:4 Changed 13 years ago by
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.
comment:5 Changed 13 years ago by
Authors: | → Robert Bradshaw |
---|---|
Reviewers: | → John Cremona |
Status: | needs_review → needs_work |
Patch applies fine to 4.4.1 and tests pass.
This functionality is badly needed!
We now have heights for points on curves defined over number_fields but no associated regulator function. I suggest that the function regulator_of_points() be moved up from ell_rational_field to ell_number_field. This tcan then be called instead of the code in lines 424-432 [line numbers are from the patched file, not the patch].
Line 439 uses a function self.height_function() which does not exist. This block needs to be replaced by something sensible. If one has a lower bound on the height of non-torsion points, then a bound on the index can be deduced from standard geometry of numbers estimates. To get such a lower bound, see papers of Cremona & Siksek (over Q) and Thongjunthug (over number fields) for an algorithm which would need to be implemented. (Not hard over Q, not much harder for totally real fields, quite a lot worse over fields with a complex embedding). Until this is done, I don't think this saturation function can allow maxprime==0.
In the rank one code: when large primes p (say, over 20!) are being tested then the division_points code will be expensive since it involves constructing the multiplication-by-p map. I would recommend using a reduction strategy here just as in the general case. To check p-saturation just find primes q such that #E(Fq) is divisible by p and then see if the reduction of P mod q is a multiple of p. This will almost always prove saturation very quickly. If it fails for several (say 5) q then try to divide P by p; else use more q, and so on. There is one theoretical drawback here: this strategy might fail if there is a rational p-isogeny. Over Q, we know which p this might happen for and I would first test for the existence of isogenies of primes degree, and use the division test (as here) for any primes that show up. Over number fields that's harder to deal with, but again we can fall back on the division test to rpove that P cannot be divided by p.
The function list_of_multiples(P,n) duplicated the generic function multiples() which I wrote for just this sort of purpose!
I don't like the loop through all linear combinations for small
primes. Even with p=2 there are curves with 24 independent points out
there and 2^24
divisions is not nice to contemplate. If you want this
short cut, do it based on the size of p^r
.
The main code with reduction etc looks fine to me (but I did not check the logic rigorously).
The gens function for E(K) when E is defined over Q and [K:Q]=2 looks fine. For a more general case we could always try using simon_two_descent (followed by saturation). Trying such an examples led me to:
sage: K.<a> = NumberField(x^2-2) sage: E = EllipticCurve([a,0]) sage: P = E(0,0) sage: P.has_finite_order() True sage: P.order() 2 sage: P.height() 0 sage: E.saturation([P], verbose=True, max_prime=5) ## infinite loop
This is caused as follows: The height matrix is [0] with det=0 but reg / min(heights) is NaN so reg / min(heights) < 1e-6 is False!. This will need fixing. At the very least, I would discard any points of finite order before doing anything else with them. Then min(heights) will never be 0.
Most of the above is easy to deal with. The hard part is computing a suitable max_prime form a lower height bound on points. I suggest that for now you make it compulsory to have a positive max_prime and add a TODO.
comment:6 follow-up: 7 Changed 13 years ago by
Thank you for all your input. self.height_function
comes from #8828, though as you suggest we could make max_prime mandatory for now (and for rank > 1 once that goes in). That's a good point about large primes in the rank one case. I found the loop through all linear combinations to be much faster in practice for small primes, but the hard coded p == 2
case was left by accident, I meant to cap that on p^r
as I did the others.
I probably won't fix and polish this up before finishing my thesis, but at the latest we should be able to get it done during the workshop at MSRI next month.
comment:7 Changed 13 years ago by
Replying to robertwb:
Thank you for all your input.
self.height_function
comes from #8828, though as you suggest we could make max_prime mandatory for now (and for rank > 1 once that goes in). That's a good point about large primes in the rank one case. I found the loop through all linear combinations to be much faster in practice for small primes, but the hard codedp == 2
case was left by accident, I meant to cap that onp^r
as I did the others.I probably won't fix and polish this up before finishing my thesis, but at the latest we should be able to get it done during the workshop at MSRI next month.
OK -- looking forward to it! I'll take a look at #8828 by then as well.
comment:8 Changed 12 years ago by
Since rwb is now busy at Google, and I want this functionality, I am now implementing the changes I suggested above!
comment:11 follow-up: 12 Changed 10 years ago by
See #12509: until we can fix the height computation, saturation cannot be carried out properly. It's still on my to-do list though.
comment:12 Changed 10 years ago by
comment:13 Changed 9 years ago by
Milestone: | sage-5.11 → sage-5.12 |
---|
comment:14 Changed 9 years ago by
Dependencies: | → #8828 |
---|
comment:15 Changed 9 years ago by
Summary: | Saturation for curves over number fields. → Saturation for MW-groups of elliptic curves over number fields. |
---|
comment:16 Changed 9 years ago by
Milestone: | sage-6.1 → sage-6.2 |
---|
comment:17 Changed 9 years ago by
Cc: | Peter Bruin added |
---|
comment:18 Changed 9 years ago by
Milestone: | sage-6.2 → sage-6.3 |
---|
comment:19 Changed 8 years ago by
Milestone: | sage-6.3 → sage-6.4 |
---|
comment:20 Changed 7 years ago by
I do not know why this was left drifting, but I really need it myself now so will look at it again, rebase on 6.8 and see what we can do. But I only have one day before a week off, so...
comment:21 Changed 7 years ago by
Authors: | Robert Bradshaw → Robert Bradshaw, John Cremona |
---|---|
Branch: | → u/cremona/8829 |
Commit: | → 0e1e35f624edb087d3fb1ddc21226fec7acfafad |
Description: | modified (diff) |
Keywords: | saturation added |
Some dependance on #8828.