elliptic curves -- implement gross-Zagier L-functions

[robertwb]

Make it so one can do:

sage: e = EllipticCurve('37a')
sage: K.<a> = QuadraticField(-40)
sage: A = K.class_group().gen(0); A
Fractional ideal class (2, -1/2*a)
sage: L = e.lseries_gross_zagier(A)  
sage: L(1)
0.000000000000000
sage: L.taylor_series(1,5)
0.000000000000000 - 5.51899839494458*z + 13.6297841350649*z^2 - 16.2292417817675*z^3 + 7.94788823722712*z^4 + O(z^5)

nobody had seen this!

CC: @williamstein @craigcitro @categorie

Component: elliptic curves

Author: William Stein, Robert Bradshaw, Craig Citro

Branch/Commit: 1c7df55

Reviewer: Chris Wuthrich

Issue created by migration from https://trac.sagemath.org/ticket/4606

[robertwb]

Attachment: 4606-gz-lseries.patch.gz

[robertwb]

comment:1

Still going to add some doctests right now, but here's the code. Note, however, the check \sum_A L_A(E,s) = L(E/K,s) fails. :(

[robertwb]

Attachment: 4606-gz-doctests.patch.gz

[craigcitro]

Attachment: trac-4606-pt3.patch.gz

[craigcitro]

comment:2

So I've attached a patch, which at least fixes a few issues. This patch fixes two definite bugs:

• In E.lseries_gross_zagier(A), you were caching the value -- no matter what value of A was getting passed in! That was an easy fix. :) Unfortunately, after fixing that, there were still issues.

• There was a subtle issue in I.quadratic_form(), for I an ideal in a quadratic number field. For instance, with just the first two patches above, taking K = QuadraticField(-40) and I the trivial element in the class group, the code returned x^2 + 40*y^2 for I.quadratic_form(). The underlying issue was one of choice of generators: you were using K.gen() where you really needed K.ring_of_integers().gen(), because you needed to know that something generated all of I over ZZ, not just QQ. I'm fairly certain that the current code is correct, but it's after 2AM, so someone should double check me.

So, now that those are fixed, we go back to the example Robert points out in the code:

sage: E = EllipticCurve('37a')
sage: K.<a> = QuadraticField(-40)
sage: A = K.class_group().gen(0)
sage: L = E.lseries_gross_zagier(A)
sage: LL = E.lseries_gross_zagier(A**2)
sage: L(2) + LL(2)
0.506799279512368

sage: E.lseries()(2) * E.quadratic_twist(-40).lseries()(2)
0.502803417587467

So we're now quite close. In particular, I wonder if there isn't rounding going on:

sage: E.lseries().taylor_series(2,5) * E.quadratic_twist(-40).lseries().taylor_series(2,5)
0.50 + 0.38*z - 0.16*z^2 + 0.0078*z^3 + 0.070*z^4 - 0.039*z^5 + O(z^6)

That definitely seems to suggest small precision to me. In any event, we're getting close:

sage: L.taylor_series(2,5) + LL.taylor_series(2,5)
0.506799279512368 + 0.360199571567893*z - 0.122141848388581*z^2 - 0.00635398874570253*z^3 + 0.0383995215484257*z^4 + O(z^5)

sage: L.taylor_series(2,5) + LL.taylor_series(2,5) - (E.lseries().taylor_series(2,5) * E.quadratic_twist(-40).lseries().taylor_series(2,5))
-0.016*z + 0.035*z^2 - 0.014*z^3 - 0.031*z^4 + O(z^5)

I'll cook up a few more examples and post what I find.

[robertwb]

comment:3

Excellent. That's looking very good. I was very tired the day I was finishing that up, so I'm glad you caught these errors.

[sagetrac-mabshoff]

comment:4

Replying to @robertwb:

Excellent. That's looking very good. I was very tired the day I was finishing that up, so I'm glad you caught these errors.

Should this ticker be "[with patch, needs review]" ?

Cheers,

Michael

[robertwb]

comment:5

No, I don't think so yet (but if Craig is happy with it, then sure).

[fchapoton]

comment:9

for the patchbots:

apply only trac_4606_gross_zagier_lseries_rebased.patch

[fchapoton]

Attachment: trac_4606_gross_zagier_lseries_rebased.patch.gz

folded all three patches and rebased on 5.12.beta5

[fchapoton]

comment:10

apply trac_4606_gross_zagier_lseries_rebased.patch

[fchapoton]

Branch: u/chapoton/4606

[fchapoton]

Commit: a356ce0

[fchapoton]

New commits:

a356ce0

Gross-Zagier L-function attached to an elliptic curve

[sagetrac-git]

Changed commit from 42ae892 to abe5e2c

[sagetrac-git]

Changed commit from abe5e2c to 0886afa

[sagetrac-git]

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

0886afa

trac #4606 more doc

[fchapoton]

comment:47

Anybody interested, please give me feed back. It seems to work.

[fchapoton]

comment:48

ping ?

[chriswuthrich]

Changed branch from u/chapoton/4606 to u/wuthrich/ticket/4606

[chriswuthrich]

Reviewer: Chris Wuthrich

[chriswuthrich]

comment:49

... pong!

I added a little bit more to the documentation. Just pointing to the article is not enough.

I tested this (merged into 6.8.beta6) and all passed.

---

New commits:

ff129c9

#4606: additional doc

[chriswuthrich]

Changed commit from 0886afa to ff129c9

[chriswuthrich]

Changed commit from ff129c9 to 0886afa

[chriswuthrich]

comment:50

oops, sorry, something went wrong. I should be able fix that..

[chriswuthrich]

Changed branch from u/wuthrich/ticket/4606 to u/chapoton/4606

[chriswuthrich]

Changed branch from u/chapoton/4606 to u/wuthrich/ticket/4606_new

[chriswuthrich]

comment:51

done and retested.

---

New commits:

1c7df55

#4606: more doc

[chriswuthrich]

Changed commit from 0886afa to 1c7df55

[chriswuthrich]

comment:52

There is one thing, I should add here, although I am in favour of putting this into sage: It is unlikely that this code will be used much. I might be wrong.

[JohnCremona]

comment:53

You may well be right, and the fact that nothing much happened to this code for 7 years tends to support that. But surely that does not matter -- far better that the code be here, properly documented and tested for the future, than that it should wither and die!

[vbraun]

Changed branch from u/wuthrich/ticket/4606_new to 1c7df55