Opened 9 years ago
Last modified 8 months ago
#9320 needs_work enhancement
Implement root numbers for elliptic curves over number fields
Reported by:  arminstraub  Owned by:  cremona 

Priority:  minor  Milestone:  
Component:  elliptic curves  Keywords:  root number 
Cc:  was, cturner, beankao, pbruin, cremona  Merged in:  
Authors:  Armin Straub  Reviewers:  
Report Upstream:  N/A  Work issues:  
Branch:  public/9320 (Commits)  Commit:  82d2ddc38912cf9d23b5783444753e060149a740 
Dependencies:  Stopgaps: 
Description (last modified by )
Root numbers for elliptic curves are currently only implemented via Pari (pari(E).ellrootno()) and only over the rational numbers.
We (Tim Dokchitser's group at Sage Days 22) intend to add the possibility to compute local and global root numbers for elliptic curves over number fields. A first patch may not fully implement the case of additive reduction over primes dividing 2 or 3.
Update: Attached is a first implementation which allows for instance:
sage: K.<a>=NumberField(x^4+2) sage: E = EllipticCurve(K, [1, a, 0, 1+a, 0]) sage: E.root_number() 1 sage: E.root_number(K.ideal(a+1)) 1
Note that the implementation needs the patches #9334 (Hilbert symbol) and #9684 ("dirty model") to be applied.
To prevent incorrect results in some cases as well as crashes, the tickets #9389 and #9417 need to be addressed.
Attachments (3)
Change History (50)
comment:1 Changed 9 years ago by
 Cc beankao added
 Description modified (diff)
 Status changed from new to needs_review
comment:2 Changed 9 years ago by
comment:3 Changed 9 years ago by
Adapted the patch to reflect the renaming of "tidy" to "reduce" following #9684.
comment:4 Changed 8 years ago by
 Status changed from needs_review to needs_work
 Work issues set to fix ReST formatting
Ticket #9334 has been merged, so this is now ready for review. Sadly it fails to apply, due to a trivial conflict with #11749. I've uploaded a rebased patch, and checked that all doctests in sage/schemes/elliptic_curves pass with this applied.
However, some (trivial but tedious) work is needed fixing the ReST formatting of the docstrings  the indentation is all over the place, and :: should only be used to introduce example code blocks.
comment:5 Changed 8 years ago by
I have tried to clean this up, but I'm not very experienced so I may have made some mistakes or missed something  could someone please have a look and help me to get it right?
comment:6 Changed 6 years ago by
apply only 9320_root_numbersrebase_docscleaned.patch
comment:7 Changed 6 years ago by
apply only 9320_root_numbersrebase_docscleaned.patch
comment:8 Changed 6 years ago by
 Work issues changed from fix ReST formatting to fix ReST formatting, coverage
seems to apply and pass all tests on 5.12.beta2
needs work to put coverage to 100%
comment:9 Changed 6 years ago by
this one just needs a little more doc (three functions need doctests)
comment:10 Changed 6 years ago by
 Branch set to u/chapoton/9320
 Commit set to e63e7b2d1cb4f4d23e2c8b527ce3c9cdf2cf0cbe
Here is a git branch
New commits:
e63e7b2  9320: Implement root numbers for elliptic curves over number fields

comment:11 Changed 6 years ago by
 Commit changed from e63e7b2d1cb4f4d23e2c8b527ce3c9cdf2cf0cbe to 87938e0bdccc397f9527b1db39b9c85006f40232
Branch pushed to git repo; I updated commit sha1. New commits:
87938e0  trac #9320 trying to add documentation, still needs work

comment:12 Changed 6 years ago by
It would be good if some expert of elliptic curve could provide correct doctests for the local root numbers at primes 2 and 3. Could you have a look, please ?
comment:13 Changed 6 years ago by
I suggest that a good source of examples would be elliptic curves over number fields where we know the associated modular form, since the root number at a bad prime should match the AtkinLehner eigenvalue. (The alternative would be to compue a whole lot of examples with Magma, but that would make me uncomfortable; nevertheless we should of course check that our results are compatible with Magma.)
There is no issue when the primes have multiplicative reduction, since then the root number is very easy being minus E.ap, i.e. depends only on whether the number of points on the reduction is p+1 or p1 (of course "p" means Norm(p) in the number field case). It's the case of additive reduction at primes dividing 2 or 3 which are harder.
Here is one taken from my thesis (see http://www.numdam.org/numdambin/search?h=nc&id=CM_1984__51_3_275_0&format=complete):
K.<i>=QuadraticField(1) E=EllipticCurve([1+i, 1+i, 0,i,0]) P2=K.ideal(i+i) E.root_number(P2) 1
which I checked with Magma. The conductor here is P2^2 * P41
.
Is this what you want? How many examples do you need?
Tables of elliptic curves over number fields do exist, and were in fact one of the topics of last week's Curves and Automorphic Forms workshop in Arizona.
comment:14 followup: ↓ 15 Changed 6 years ago by
One example would be enough, I think if it is bad at both 2 and 3. Maybe one can just use the one above as "indirect doctest" ?
Do you really mean K.ideal(i+i)
?
comment:15 in reply to: ↑ 14 Changed 6 years ago by
Replying to chapoton:
One example would be enough, I think if it is bad at both 2 and 3. Maybe one can just use the one above as "indirect doctest" ?
I'll find another example at 3. I don't know the code but I expect that it also depends on factors such as the ramification degree of the prime, so I'll give an example over Q(sqrt3) with additive reduction at 3. Wait for it...
Do you really mean
K.ideal(i+i)
?
Yes
comment:16 Changed 6 years ago by
An example which is additive at 3:
sage: K.<r> = NumberField(x^2x+1) sage: E = EllipticCurve([r1,r,1,r1,1]) sage: P3 = K.ideal(2*r1) sage: assert P3.is_prime() and P3.norm()==3 sage: assert E.has_additive_reduction(P3) sage: assert P.root_number(P3)==1 ## not implemented
Here the value is taken from page 115 of my thesis, and agrees with Magma's value.
For additional testing we could put in an 'algorithm' parameter which could be 'magma' and then (of course) only work when Magma is installed, so you could have optional doctests conditional on Magma of the form
assert E.root_number(P) == E.root_number(P, algorithm='magma')
comment:17 Changed 6 years ago by
 Commit changed from 87938e0bdccc397f9527b1db39b9c85006f40232 to a0462dfb7aae331e112ed2979d7fffbd99319fc4
Branch pushed to git repo; I updated commit sha1. New commits:
a0462df  trac #9320 work done on code and doc, but still one failing doctest

comment:18 Changed 6 years ago by
 Commit changed from a0462dfb7aae331e112ed2979d7fffbd99319fc4 to d40bf637722cd2cb68b1ce0310290bbed2f7a1f5
Branch pushed to git repo; I updated commit sha1. New commits:
d40bf63  trac #9320 doctest is indirect

comment:19 Changed 6 years ago by
Thanks a lot for the examples. I have done what I could for this code, but there is now a failing doctest for the example you provided which has additive reduction at 2.
And I had to replace K.ideal(i+i)
by K.ideal(1+i)
...
Not being in my field of expertise, I will now leave this ticket in more able hands.
comment:20 Changed 5 years ago by
I had a quick look at this and there are indeed major issues with the local root number at places above 2 when the reduction is additive, potentially good. That is in the function _root_number_local_2
in ell_number_field.py
.
With the current commit above we get an error for
sage: K.<i> = QuadraticField(1) sage: E = EllipticCurve([1+i, 1+i, 0, i, 0]) sage: P2 = K.ideal(1+i) sage: E.root_number(P2)
because line 1966 tries to create and extension with a reducible polynomial K2 = K.extension(t**2+E.a2()*t+E.a4(), 'a2')
. I guess  but it would take me some time to read the sources  that K2 should just be K in this case.
However there are other issues. For instance
sage: E = EllipticCurve([1+i,0,0,0,i]) sage: E._root_number_local_2(P2)
goes "bang".
I would think the code without the very bad case at places above 2 is ok and that could already go in. So I see two possible futures for this ticket:
 Someone fixes these issues above 2 or
 We raise an
NotImplementedError
if the reduction is additive and potentially good at a place above 2 and open a new ticket for fixing the problem.
I can help with 2), but I won't have time to do 1). But maybe there is someone out there who would.
comment:21 Changed 5 years ago by
It all started a long time ago, at the Sage Days & summer school at MSRI. Perhaps we good get Tim Dokchitser, who led the original project (and who developed the background theory, I think) involved? He works mainly in Magma, but back in 2010 he certainly was involved with this.
comment:22 Changed 5 years ago by
 Cc pbruin added
comment:23 Changed 4 years ago by
 Commit changed from d40bf637722cd2cb68b1ce0310290bbed2f7a1f5 to b162b900fb9213f5fde451d69211fcf1f5ad25bf
Branch pushed to git repo; I updated commit sha1. New commits:
b162b90  Merge branch 'u/chapoton/9320' of trac.sagemath.org:sage into 9320

comment:24 Changed 4 years ago by
 Commit changed from b162b900fb9213f5fde451d69211fcf1f5ad25bf to 62975879c068b50731fe3faca2280b68acaa5b71
Branch pushed to git repo; I updated commit sha1. New commits:
6297587  trac #9320 add missing import

comment:25 Changed 4 years ago by
 Commit changed from 62975879c068b50731fe3faca2280b68acaa5b71 to fee78bb780730a9089eb7ed776d71d7d5e515263
Branch pushed to git repo; I updated commit sha1. New commits:
fee78bb  Merge branch 'u/chapoton/9320' into 6.10.b1

comment:26 Changed 4 years ago by
comment:27 Changed 4 years ago by
comment:28 Changed 4 years ago by
Sorry, I reverted that. Tim did not touch much of the code himself as far as I can remember that long long time ago in Berkley. I doubt he would want to vouch for this code by himself...
comment:30 Changed 4 years ago by
What is the issue ? Why can't you leave the original string in there ?
comment:31 Changed 4 years ago by
Because "Tim Dokchitser and group (Sage Days 22)" isn't an actual person.
comment:32 Changed 4 years ago by
But it was a collaborative effort. The wiki for the sage days lists the participants in this group as "Armin, Charlie, Hatice, Christ, Lola, Robert Miller, Thilina, M. Tip, Robert Bradshaw " I am not sure who actually did the coding and I don't remember all full names. So the original string was probably the closest to who the author was. Otherwise set it to Armin Straub, who did the original uploading onto this trac ticket.
comment:33 Changed 4 years ago by
comment:34 Changed 4 years ago by
 Work issues fix ReST formatting, coverage deleted
there are some failing doctests..
comment:35 Changed 4 years ago by
 Commit changed from fee78bb780730a9089eb7ed776d71d7d5e515263 to b4f448ced7e76f9285106f40a5054b1258f19851
Branch pushed to git repo; I updated commit sha1. New commits:
b4f448c  Merge branch 'u/chapoton/9320' into 7.1.b5

comment:36 Changed 4 years ago by
 Commit changed from b4f448ced7e76f9285106f40a5054b1258f19851 to bd9aa7fdcaf2ac0857e42820281bd9090c5d2be0
comment:37 Changed 4 years ago by
 Commit changed from bd9aa7fdcaf2ac0857e42820281bd9090c5d2be0 to 28cb70ca84dc378dcbd446a53fc671c7e4efdcf3
Branch pushed to git repo; I updated commit sha1. New commits:
28cb70c  trac #9320 remove the algo keyword and the duplicate code

comment:38 Changed 4 years ago by
 Cc cremona added
Could some expert in elliptic curves have a look at the last failing doctest, please ?
comment:39 Changed 4 years ago by
I can simply repeat my comment:20 above. This is a genuine bug.
My favourite solution is still to raise non implemented warnings for the cases this code does not do currently.
Before accepting this ticket, the reviewer will have to do lots of comparison with magma. The code in magma will have plenty less bugs than this one as the Dokchitsers have worked a lot on that code.
comment:40 Changed 4 years ago by
 Commit changed from 28cb70ca84dc378dcbd446a53fc671c7e4efdcf3 to 7a131d0b685656947d273c4e9de30bab69408935
Branch pushed to git repo; I updated commit sha1. New commits:
7a131d0  trac #9320 trying to fix the issues for potential good reduction over 2

comment:41 Changed 4 years ago by
I tried to fix the problem, but only with partial success. It seems that at some point the code assumes vanishing of coeff a1 in long Weierstrass equation, and I cannot see why.
comment:42 Changed 3 years ago by
 Commit changed from 7a131d0b685656947d273c4e9de30bab69408935 to 60e9a78c6facda896d1ec681c7b429d3b832bdc8
Branch pushed to git repo; I updated commit sha1. New commits:
60e9a78  Merge branch 'u/chapoton/9320' in 7.3.rc0

comment:43 Changed 2 years ago by
 Commit changed from 60e9a78c6facda896d1ec681c7b429d3b832bdc8 to 66f60b95d17439eb53b8dea2a9dde34963b939bd
Branch pushed to git repo; I updated commit sha1. New commits:
66f60b9  Merge branch 'u/chapoton/9320' in 8.1.b5

comment:44 Changed 21 months ago by
 Commit changed from 66f60b95d17439eb53b8dea2a9dde34963b939bd to c34b3080820198115a79dee956550acc5a495f34
Branch pushed to git repo; I updated commit sha1. New commits:
c34b308  Merge branch 'u/chapoton/9320' in 8.2.b2

comment:45 Changed 8 months ago by
 Branch changed from u/chapoton/9320 to public/9320
 Commit changed from c34b3080820198115a79dee956550acc5a495f34 to 223f82090ca704dd4ae71715eb661adf727b6522
comment:46 Changed 8 months ago by
 Commit changed from 223f82090ca704dd4ae71715eb661adf727b6522 to 82d2ddc38912cf9d23b5783444753e060149a740
Branch pushed to git repo; I updated commit sha1. New commits:
82d2ddc  fix indentation error in previous commit

comment:47 Changed 8 months ago by
After merging with 8.6.beta1 I ran the tests, and there is one failure in line 2294 of ell_number_field where a local root number at a ramified prime above 2 where there is additive reduction is computed incorrectly.
I checked with Magma that 1 is the correct value. This curve is http://beta.lmfdb.org/EllipticCurve/2.0.4.1/164.2/a/2 and has associated Bianchi newform http://beta.lmfdb.org/ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/164.2/a/ where you can see that the AtkinLehner eigenvalue at 1+i is 1, giving a second independent evaluation.
I don't think it is possible to review this until the ticket it depends on (#9334) which needs work has been fixed.