Fwd: Table of Elliptic Curves over Q(sqrt(5))
William Stein
---------- Forwarded message ----------
From: Noam Elkies
Date: Friday, August 12, 2011
Subject: Table of Elliptic Curves over Q(sqrt(5))
To: wst...@gmail.com
>> Thanks, and congratulations! I also saw your "Wall" post on Facebook,
>> which said one curve was found today with help from Tom Fisher.
> True -- that has norm conductor 1476 < 1831 = norm conductor of
> first rank 2 curve < 1856 = where we have a missing curve.
Ah, I see.
I suppose you already tried to obtain this missing curve via quadratic
twists of known curves of this or smaller conductor? Do you have
a long enough aplist to detect isogenies or determine the Galois
structure of E[2] (and maybe even E[3]), which would considerably
narrow down the search?
>> Also, are there by now enough modularity theorems in this setting
>> to be sure that you have *all* curves up to the first curve of rank 2,
>> rather than just all modular ones?
> No, definitely not. I have a recent email from Richard Taylor
> sketching out a strategy for proving modularity over Q(sqrt(5)), which
> still has serious gaps in it.
I see; I figured that if modularity had been proved in this setting I
might have heard of it, but since Wiles-Taylor there have been so many
modularity results that it's not easy to keep track of the state of the art.
> In the README.txt file at the page, I state our assumption of modularity.
Ah, sorry, I see this now; I entirely skipped that line when reading
through it the first time.
>> Also, I note that there's no isogeny information [...]
> Yes, we have this. The students just haven't drawn all the graphs and
> put them into the pdf. But we do have the isogeny diagram
> information, with proof that they are complete.
OK, I'll be patient.
> Thanks for your help and feedback!
You're welcome,
--NDE
--
William Stein
Professor of Mathematics
University of Washington
http://wstein.org
William Stein
From: Noam Elkies <elk...@math.harvard.edu>
Date: Sun, Aug 14, 2011 at 9:16 AM
Subject: Re: Table of Elliptic Curves over Q(sqrt(5))
To: wst...@gmail.com
john.c...@gmail.com, jvo...@gmail.com,
uwntr...@googlegroups.com
Dear William et al., (*)
A few more comments after looking some more at the neat new table:
-- I note some new phenomena not seen for elliptic curves over Q,
such as a 15-torsion point (on 100a:B1, which is the Antwerp curve
[1,1,1,-3,1] of conductor 50 over Q), and isogeny classes with
as many as 10 curves, starting with 45a:A1-10. These are actually
related, because one of the curves in this class is the Antwerp curve
[1,1,1,0,0] of conductor 15 over Q, which is none other than the
modular curve X_1(15), and the fact that it has a torsion point over
Q(sqrt(5)) that's not rational over Q means that there's a curve over
Q(sqrt(5)) with 15-torsion. [The Q-rational points are cusps; there
are four new points, but only one new curve because they're in one
orbit under the diamond operators; the curve is defined over Q because
the 4-isogeny to X_0(15) takes the new torsion point to a rational one.]
The 10-curve isogeny classes, in turn, come from rational points on
X_0(32), which is an elliptic curve of rank 0 over Q but whose quadratic
twist by Q(sqrt(5)) has rank 1 and thus infinitely many points. The
"remarks on isogenies" preceding the Antwerp tables (and apparently not
included in the scans at modular.math.washington.edu/Tables/antwerp :-( )
list the other 10 curves X_0(N) of genus 1; most of them still have
rank zero over Q(sqrt(5)), but there are two more cases with rank 1,
one of which produces another of the novel isogeny structures in your
table, the 27-isogenies first seen at 76a:B1-4 and 76a:B1-4. The remaining
case where X_0(N) has rank 1 seems to be just outside the range of your
table: the smallest conductor I found has norm 58^2, with multiplicative
reduction at the prime above 2 and additive reduction at one of the primes
above 29. (When I first saw this 29 appearing, I thought I might have also
run across your missing curve of conductor norm 1856; but no, that one's
additive at 2 and multiplicative at a prime above 29.) I'm not specifying
the curves here, in case you might want to use them to test your curve- and
isogeny-finding routines; there should be two exotic isogeny classes at this
level for each of the primes above 29, related by quadratic twist.
I also mentioned already a few examples beyond your tables that feature
isogenies parametrized by Q(sqrt(5))-rational points on curves X_0(N)
of genus > 1; one example is [0, 0, 1, -322 - 46*phi, -3571 - 2645*phi]
(yes, there's also a 5-isogeny).
-- Like the Tingley (Antwerp) and Cremona tables, yours list local invariants
s, ord(Delta), ordminus(j), c_p, and Kodaira at each prime factor of N.
Unlike the earlier tables, yours seems to list this information in reverse
order when there's more than one bad prime! (At least that's the case for
a few cases with multiplicative reduction that I tried.) Assuming that
this is unintentional, I hope it's easy to fix.
-- Apropos prime notation, I understand why you need suffixes a,b for primes
and a,b,... for conductor level, buth I don't know what convention you
chose for the ordering, nor why the unique prime above 5 gets a suffix
(unlike the inert primes -- maybe that one ramified prime should have
a unique symbol, like \sqrt{5}). In any case, there should be a small
auxiliary table where one can find how Galois conjugation acts on the curves.
-- Along the same lines: what is the meaning of "aplist [0,4,-1,1,1,6,...]"
(http://wstein.org/home/wstein/reu/2011/final/README.txt) -- that is, in what
order are you listing the primes p -- and is it easy to generate more a_p's
if needed to determine uniquely the Galois-module structure on E[2] etc.?
In principle this should not be needed because one could just try each of
the relatively small number of possibilities that are unramified away from
2 and 29a, but if the a_p information is already available it might as well
be used.
NDE
(*) including Jordan Ellenberg, in reply to his blog post
William Stein
From: Dino J Lorenzini <lore...@uga.edu>
Date: Sun, Aug 14, 2011 at 8:34 PM
Subject: RE: Table of Elliptic Curves over Q(sqrt(5))
To: William Stein <wst...@gmail.com>
Hi William,
Great job, such a table should become useful.
By the way, would it be possible to not just give the order of the
torsion, but the full group structure,
for example by introducing the notation (2,4) when the group has order
8 but is Z/2 x Z/4
I think that it would be useful and probably not much more expensive?
Best,
Dino
________________________________________
From: Number Theory List [NMBR...@LISTSERV.NODAK.EDU] on behalf of
William Stein [wst...@gmail.com]
Sent: Saturday, August 13, 2011 3:04 AM
To: NMBR...@LISTSERV.NODAK.EDU
Subject: Table of Elliptic Curves over Q(sqrt(5))
Hello Number Theorists,
This summer I organized an REU that made a comprehensive table of
elliptic curves over Q(sqrt(5)) similar to Cremona's highly
influential tables over Q, which may be of interest to the number
theory community:
http://wstein.org/home/wstein/reu/2011/final/
We intend to write a paper outlining our techniques for ANTS 10
(http://math.ucsd.edu/~kedlaya/ants10/).
-- William
R. Andrew Ohana
-- Apropos prime notation, I understand why you need suffixes a,b for primes
and a,b,... for conductor level, buth I don't know what convention you
chose for the ordering, nor why the unique prime above 5 gets a suffix
(unlike the inert primes -- maybe that one ramified prime should have
a unique symbol, like \sqrt{5}). In any case, there should be a small
auxiliary table where one can find how Galois conjugation acts on the curves.
-- Along the same lines: what is the meaning of "aplist [0,4,-1,1,1,6,...]"
(http://wstein.org/home/wstein/reu/2011/final/README.txt) -- that is, in what
order are you listing the primes p -- and is it easy to generate more a_p's
if needed to determine uniquely the Galois-module structure on E[2] etc.?
In principle this should not be needed because one could just try each of
the relatively small number of possibilities that are unramified away from
2 and 29a, but if the a_p information is already available it might as well
be used.
NDE
(*) including Jordan Ellenberg, in reply to his blog post
http://quomodocumque.wordpress.com/2011/08/13/cremona-tables-for-elliptic-curves-over-qsqrt5-by-william-stein-and-his-reu/
I think it may have been good for William to not have called these 'final' tables, but rather saturated tables -- since clearly we have more kinks to work out still, now that most of the hard work is finished.
R. Andrew Ohana
On Aug 16, 2011 8:04 AM, "Noam Elkies" <elk...@math.harvard.edu> wrote:
>
> "R. Andrew Ohana" <andrew...@gmail.com> writes:
>
> > [I had written:]
>
> >> -- Like the Tingley (Antwerp) and Cremona tables, yours list
> >> local invariants [...] at each prime factor of N. Unlike
> >> the earlier tables, yours seems to list this information in reverse
> >> order when there's more than one bad prime! (At least that's
> >> the case for a few cases with multiplicative reduction that I tried.)
> >> Assuming that this is unintentional, I hope it's easy to fix.
>
> > Yes this is unintentional, and in fact they aren't even necessarily
> > ordered in reverse! [...]
>
> Just as well I didn't try more examples and then get even more confused...
>
> > Hopefully I can get this fixed soon, however, I'm currently working
> > on including the isogeny graphs included, as well as potentially
> > reordering the tables (in an effort to make sure that our ordering
> > generalizes well).
>
> OK, as long as this is on your agenda.
>
> >> -- Apropos prime notation, [..] I don't know what convention you
> >> chose for the ordering [of ideals of a given norm], nor why the
> >> unique prime above 5 gets a suffix (unlike the inert primes --
> >> maybe that one ramified prime should have a unique symbol,
> >> like \sqrt{5}). In any case, there should be a small auxiliary table
> >> where one can find how Galois conjugation acts on the curves.
> >
> > Currently the ordering on ideals is rather complicated:
>
> > 1) we first order by norm
> > 2) we then order by the min{ tr(alpha^2) : alpha^2 }
>
> Strange -- though it does explain why 961a is 31, preceding
> the squares of the two primes of norm 31 that appear as 961b and 961c.
>
> > 3) finally we order by the lift in [0,p-1] of image of phi into O_K/J,
> > where J = I / < I =E2=88=A9 Z >
>
> What is the symbol that appears above as "=E2=88=A9"? It appears
> even stranger in the HTML. (In general, better to stick with ASCII only
> for e-mail -- for most purposes "rich text" and HTML e-mail are a bug,
> not a feature.)-:
>
Sorry about that, < I =E2=88=A9 Z > should be the ideal generated by I intersect Z.
> > we decided to denote primes that do not have prime norm (the inert ones)
> > without a suffix so that the reader is immediately aware for which primes
> > the norm is square.
>
> Well I guess most readers will already appreciate the nature of 5...
>
> >> -- Along the same lines: what is the meaning of "aplist [0,4,-1,1,1,6,...]"
> >> (http://wstein.org/home/wstein/reu/2011/final/README.txt) -- that is,
> >> in what order are you listing the primes p -- and is it easy to generate
> >> more a_p's if needed [...]
>
> > Primes are ordered as mentioned above, and yes more a_p have been computed
>
> So are these the primes of norm 4, 5, 9, 11, 11, 19, ... or did you already
> omit 4 because it's a prime of bad reduction? In the former case, it so
> happens that the two primes of norm 11 have the same trace and one can ask
> whether this persists (if so you have a Q-curve).
It should be the first case, although we did encounter issues with the ap values being incorrect for primes of bad reduction in another curve that was missing.
>
> > -- I think up to norm 50,000 (William correct me if I'm wrong).
>
> Wow!
>
> > I think it may have been good for William to not have called these
> > 'final' tables, but rather saturated tables -- since clearly we have
> > more kinks to work out still, now that most of the hard work is finished.
>
> That's what I expected, since there's clearly a blank space at the end of
> each row where isogeny information should go. Are you literally doing
> isogeny graphs a la Antwerp, or just tabulating each curve's isogenies
> as in Cremona's table? Some of the more complicated graphs might be
> hard to accommodate in the space available...
Hopefully we can make graphs, a la Antwerp, as they appear to not get to be too complicated up to our bound. If we can't fit it, we'll probably tabulate like Cremona.
>
> Thanks for the further information,
> --NDE
--
Andrew