Implement root numbers for elliptic curves over number fields

[arminstraub]

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.

CC: @williamstein @sagetrac-cturner @sagetrac-beankao @pjbruin @JohnCremona

Component: elliptic curves

Keywords: root number

Author: Armin Straub

Branch/Commit: public/9320 @ 82d2ddc

Issue created by migration from https://trac.sagemath.org/ticket/9320

[arminstraub]

Author: Tim Dokchitser and group (Sage Days 22)

[JohnCremona]

comment:2

I don't think it is possible to review this until the ticket it depends on (#9334) which needs work has been fixed.

[arminstraub]

Attachment: 9320_root_numbers.patch.gz

requires #9334 and #9684

[arminstraub]

comment:3

Adapted the patch to reflect the renaming of "tidy" to "reduce" following #9684.

[loefflerd]

Rebased to 4.8.alpha5

[loefflerd]

Work Issues: fix ReST formatting

[loefflerd]

comment:4

Attachment: 9320_root_numbers-rebase.patch.gz

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.

[sagetrac-cturner]

Attachment: 9320_root_numbers-rebase_docscleaned.patch.gz

docstrings improved

[sagetrac-cturner]

comment:5

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?

[fchapoton]

comment:6

apply only 9320_root_numbers-rebase_docscleaned.patch

[fchapoton]

comment:7

apply only 9320_root_numbers-rebase_docscleaned.patch

[fchapoton]

Changed work issues from fix ReST formatting to fix ReST formatting, coverage

[fchapoton]

comment:8

seems to apply and pass all tests on 5.12.beta2

needs work to put coverage to 100%

[fchapoton]

comment:9

this one just needs a little more doc (three functions need doctests)

[fchapoton]

comment:10

Here is a git branch

---

New commits:

e63e7b2

9320: Implement root numbers for elliptic curves over number fields

[fchapoton]

Branch: u/chapoton/9320

[fchapoton]

Commit: e63e7b2

[sagetrac-git]

Changed commit from e63e7b2 to 87938e0

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

87938e0

trac #9320 trying to add documentation, still needs work

[fchapoton]

comment:12

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 ?

[JohnCremona]

comment:13

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 Atkin-Lehner 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 p-1 (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/numdam-bin/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.

[fchapoton]

comment:14

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) ?

[chriswuthrich]

comment:30

What is the issue ? Why can't you leave the original string in there ?

[jdemeyer]

comment:31

Because "Tim Dokchitser and group (Sage Days 22)" isn't an actual person.

[chriswuthrich]

comment:32

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.

[jdemeyer]

Changed author from ??? to Armin Straub

[fchapoton]

Changed work issues from fix ReST formatting, coverage to none

[fchapoton]

comment:34

there are some failing doctests..

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

b4f448c

Merge branch 'u/chapoton/9320' into 7.1.b5

[sagetrac-git]

Changed commit from fee78bb to b4f448c

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

3e3f4aa	Merge branch 'u/chapoton/9320' into 7.1.b6
bd9aa7f	trac #9320 failing doctest

[sagetrac-git]

Changed commit from b4f448c to bd9aa7f

[sagetrac-git]

Changed commit from bd9aa7f to 28cb70c

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

28cb70c

trac #9320 remove the algo keyword and the duplicate code

[fchapoton]

comment:38

Could some expert in elliptic curves have a look at the last failing doctest, please ?

[chriswuthrich]

comment:39

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.

[sagetrac-git]

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

[sagetrac-git]

Changed commit from 28cb70c to 7a131d0

[fchapoton]

comment:41

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.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

60e9a78

Merge branch 'u/chapoton/9320' in 7.3.rc0

[sagetrac-git]

Changed commit from 7a131d0 to 60e9a78

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

66f60b9

Merge branch 'u/chapoton/9320' in 8.1.b5

[sagetrac-git]

Changed commit from 60e9a78 to 66f60b9

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

c34b308

Merge branch 'u/chapoton/9320' in 8.2.b2

[sagetrac-git]

Changed commit from 66f60b9 to c34b308

[JohnCremona]

comment:45

Merged with 8.6.beta1

---

New commits:

223f820

Merge branch 'develop' into 9320

[JohnCremona]

Changed commit from c34b308 to 223f820

[JohnCremona]

Changed branch from u/chapoton/9320 to public/9320

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

82d2ddc

fix indentation error in previous commit

[sagetrac-git]

Changed commit from 223f820 to 82d2ddc

[JohnCremona]

comment:47

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 Atkin-Lehner eigenvalue at 1+i is -1, giving a second independent evaluation.