projects

[William Stein]

Hi,

I made a list of 14 possible projects for Sage Days 18:

http://wiki.sagemath.org/dayscambridge2/sprints

Obviously only a subset of them will really happen -- perhaps only 4 or 5?
If you have any additional ideas, or would like to somehow contribute to
one of the listed projects, please feel free to edit the wiki page or just
respond to this email.

William

--
William Stein
Associate Professor of Mathematics
University of Washington
http://wstein.org

[dr...@math.mit.edu]

Hi William,

I would definitely be motivated to work on the table of Galois images
project that you suggested in your list. I am currently rerunning my
previous computations on the Stein-Watkins database using an improved
version of the algorithm (just for the mod ell case at the moment, I still
want to tweak the mod ell^k code some more). It would be great to get all
this data organized and accessible in a useful form, especially while
everything is fresh in my mind.

Drew

[Jared Weinstein]

This might be idle blather, but I've been thinking about Kolyvagin classes and I'm curious about the following.  On the one hand, one of Kolyvagin's suite of conjectures is that the classes he constructs out of Heegner points ought to generate the entirety of Sha(E/K).  On the other hand each element of Sha(E/K) is "visible" in some other abelian variety, in the sense of Mazur (see for instance http://www.williamstein.org/home/was/www/home/wstein/www/papers/visibility_of_sha/).  I wonder if the Kolyvagin classes d(n) contributing to Sha become visible in abelian varieties in some *systematic* way;  ie, in a modular Jacobian of possibly higher level.  There are already some examples out there of computation with visibility and computation with Kolyvagin classes, so maybe we can gather some data.

Looking forward to next week,

Jared

[William Stein]

On Sat, Nov 28, 2009 at 1:50 PM, Jared Weinstein
<jaredsw...@gmail.com> wrote:
> This might be idle blather, but I've been thinking about Kolyvagin classes
> and I'm curious about the following.  On the one hand, one of Kolyvagin's
> suite of conjectures is that the classes he constructs out of Heegner points
> ought to generate the entirety of Sha(E/K).  On the other hand each element
> of Sha(E/K) is "visible" in some other abelian variety, in the sense of
> Mazur (see for instance
> http://www.williamstein.org/home/was/www/home/wstein/www/papers/visibility_of_sha/).
> I wonder if the Kolyvagin classes d(n) contributing to Sha become visible in
> abelian varieties in some *systematic* way;  ie, in a modular Jacobian of
> possibly higher level.  There are already some examples out there of
> computation with visibility and computation with Kolyvagin classes, so maybe
> we can gather some data.

Thanks. I've added your project suggestion here:

http://wiki.sagemath.org/dayscambridge2/sprints#VisibilityofKolyvaginClasses

and, yes, I've thought about this question some, as has Dimitar Jetchev.

William

[William Stein]

On Sat, Nov 28, 2009 at 7:35 PM, Noam Elkies <elk...@math.harvard.edu> wrote:
>> I made a list of 14 possible projects for Sage Days 18:
>
>>    http://wiki.sagemath.org/dayscambridge2/sprints
>

> I read there:
>
> ::: Implement computation of the 3-Selmer rank of an elliptic curve over Q
>
> Remember that my student Jeechul Woo has already done a big part of the
> special but important case that the curve E has a rational 3-torsion point P:
> he gives a formula for a matrix over the 3-element field whose row and
> column kernels are the Selmer groups for descent by the 3-isogenies between
> E and E/<P>.

Thanks! Did he write this up anywhere?

>
> ::: Compute a Heegner point on the Jacobian of a genus 2 curve
>
> I did this for a few special cases about 15 years ago: I find in
> my files from June 1993

Awesome! Is there any chance you could email me that file? It would
be nice for us to replicate that computation then go (much) further.

> a computation of some CM points on the
> modular curve X_0(67)/w of genus 2.  I found back then that the
> curve has Weierstrass form
>
>  y^2 = t^6 + 2t^5 + t^4 - 2t^3 + 2t^2 - 4t + 1
>
> -- or equivalently  u^2 - (t^3+t^2+1)u + t^3+t = 0,  with good reduction at 2
> and smaller coefficients -- and noted that
>
> " The rational points on this curve include the cusp (t,u)=(inf,1)
> " and 9 CM points, with t-coordinates as follows:
> "
> " |D|:  3 7 8  11 12 27 28 43 67
> " ---+--------------------------
> "  t : -1 0 0 inf  1 -1 -2  1 -2
> "
> " These are presumably the only rational points on this curve;
> " at any rate there are no others with t=m/n and |m|,|n| <= 100.
>
> These days ratpoints extends the search to height 10^5 (with the same
> results) in a few seconds, and the genus-2 experts can probably prove
> that this list of rational points is complete.  I even did some examples
> along the same lines with curves X_0(p)/w of genus 3; e.g. my 1998 paper
> "Elliptic and modular curves over finite fields and related computational
> issues" gives an explicit plane quartic model for the p=239 case and locates
> the cusp and the CM points of discriminants -7, -19, -28, and -43.
>
> So you probably want to do this for a genus-2 curve C whose Jacobian
> has real multiplication, and is thus an isogeny factor of some J_0(N),
> but for which C is not itself dominated by that X_0(N).

Yes, that's exactly right. Many thanks for your input (especially
since you won't be around for the actual workshop).

William

[Noam Elkies]

>> Remember that my student Jeechul Woo has already done a big part of
>> the special but important case that the curve E has a rational

>> 3-torsion point P: [...]

> Thanks! Did he write this up anywhere?

Not yet -- it's only a month or two old, and we're hoping he can do the
full 3-descent before he must hand in his thesis. You might remember
that I suggested you invite him to talk about this at Sage Days 22
in MSRI next June.

>> ::: Compute a Heegner point on the Jacobian of a genus 2 curve

>> I did this for a few special cases about 15 years ago: I find in
>> my files from June 1993

> Awesome! Is there any chance you could email me that file? It would
> be nice for us to replicate that computation then go (much) further.

See the appended copy. The first step is much easier now that you've
tabulated the modular forms to many more coefficients than are
necessary for this purpose (since the level N is prime it could also
have been done using Atkin's mod-N technique). I have similar files
for levels 73, 103, 167, 191 (the last two are even older, from 1991,
and the case of X_0(191)/w is also in the paper I cited in my previous
e-mail).

It seems that the first case where the Jacobian of X_0(p)/w has an
isogeny factor of dimension 2 that isn't the whole curve is p=223
(http://modular.math.washington.edu/Tables/antwerp/table5/small_140.jpg).
I found this factor also on your website at
<http://modular.math.washington.edu/tables/arith_of_factors/data/223>
but I can't seem to retrieve such information from any of the links at
<http://modular.math.washington.edu/Tables> -- where would I find this?

> Yes, that's exactly right. Many thanks for your input (especially
> since you won't be around for the actual workshop).

You're welcome. I might be able to drop in Saturday -- it's the
educational day (and I also want to spend some time on the Putnam
problems) but I figure you'll still be there.

NDE

6767676767676767676767676767676767676767676767676767676767676767676

[June 2, 1993]

Let A,B be the dual quadratic forms

[20, -3, 4, -1; -3, 8, 1, -1; 4, 1, 18, 0; -1, -1, 0, 2]

and

[4, 2, -1, 3; 2, 10, -1, 6; -1, -1, 4, -1; 3, 6, -1, 38]

of discriminant 67^2. Let phi be the w-invariant cuspform of weight 2
for X_0(67): phi = (theta_A - theta_B) / 2. Then the w-invariant
cuspforms are generated by

phi2 = 2*phi + T2(phi) = [0, 1,-1,-3, 0, 0, 3, 4, 3, ...] ,
phi1 = phi + 2*phi2 = [1, 0,-3,-3,-3, 1, 4, 3, 5, ...] .

(The normalized eigenforms are phi1 + k*phi2 where k^2+3k+1=0.)

Let t = phi1/phi2 - 1 = [1,_0_,1,1,1,2,2,1,5,3,4,7,4,...]
and y = (-q dt/dq)/phi2 = [1,2,3,_4_,10,11,24,35,...]

Then y^2 = t^6+2t^5+t^4-2t^3+2t^2-4t+1. [This polynomial happens
to have Galois group the transitive A5.] Even nicer is to take
u = (t^3+t^2+1-y)/2 = [_1_,-1,2,-1,-2,4,-4,1,7,...] when our
equation for X*(67) becomes u^2 - (t^3+t^2+1)u + t^3+t = 0.