Table of Elliptic Curves over Q(sqrt(5))
William Stein
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
--
William Stein
Professor of Mathematics
University of Washington
http://wstein.org
William Stein
> Hi William,
>
> You write:
>
>> 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/
>
> 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.
> Yet
> the README file http://wstein.org/home/wstein/reu/2011/final/README.txt
> says one class of norm 1856 is still missing.
That's true.
> 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. In the README.txt file at the page, I
state our assumption of modularity.
> Also, I note that there's no isogeny information in the
> PDF table at <http://wstein.org/home/wstein/reu/2011/final/table.pdf>
> (which goes only up to 1831); do you already have this information or
> is it not yet computed?
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.
> Scanning down the table, the first cases of rank 1 seem to be the
> "a" curves of conductor norm 199, and it's a bit surprising that here
> there's nontrivial torsion -- over Q there's no example known (and
> likely none expected) of a record curve for some positive rank
> that has nontrivial isogenies, let alone torsion.
Yes.
> I had to go down
> several more pages and rank-1 curves before finding the first example
> of rank 1 with trivial isogenies (conductor norm 229). I guess
> the torsion etc. for 199 were known because that curve was already
> found by Lassina Dembele and noted in your talk
> <modular.math.washington.edu/talks/2011-02-11-sqrt5/slides/sqrt5.pdf>
> but I never checked for that.
Yes.
Thanks for your help and feedback!
-- William
>
> NDE