Re: today
William Stein
> On Tue, 5 May 2009, William Stein wrote:
>> Tom B.: We're meeting for some reason at 2:00 today. I can't remember
>> why.
>
> Aly and Sourav will be there too. We want to discuss projects for your 583
> class.
Here are some project ideas:
COMPUTATIONAL PROJECTS:
* Compute some Heegner points on elliptic curves over QQ.
* Compute a Heegner point on the rank 0 abelian variety J_0(23) over
C (as an element of CC^2/Period lattice).
* Compute the *height* of a single Heegner points using Gross-Zagier
formula on the *rank 2* abelian variety A (of dimension 2) that is a
factor of J_0(67). To apply Gross-Zagier will require knowing the
volume of the period lattice, which currently requires Magma (but see
below).
* Compute some sort of multigraph or other combinatorial structure
associated to intersections of all subsets of {A_f : f is a newform of
level dividing N}. Due this for N < 100, say. There is code in Sage
already to compute the intersections, so just need to run it, store
the results, decide what they are combinatorially, make
tables/pictures.
* Compute the *period lattice* associated to a newform of degree > 1.
There is no code in Sage to do this yet, but there is code in Magma
(that I wrote), and the algorithm is described in my thesis and
Cremona's book. It involves summing some infinite series numerically.
Giving error bounds (hence provable correct precision) would be
pretty cool, since I never did that.
* Search for an abelian variety A_f^{\dual} in J_0(N) with End(A_f)
=/= O_f. "Study" that A_f if you find one. Otherwise, remark that
you didn't find any for N up to some bound.
* Implement computing isogenies between elliptic curves (Dan Shumow
is working on this).
THEORETICAL PROJECTS:
* Write a paper on the possibilities for End(A_f/Q) and
End(A_f/Qbar). You would (1) prove that for any simple abelian
variety A over Q, we have that End(A/Q) is a finitely generated
abelian group of rank at most dim(A), and (2) state (without proof)
what Ribet proves about End(A_f/Qbar).
* Write a paper explaining how X_0(N) has a complex structure.
I.e., explain carefully how the orbits for the extended upper half
plane under the action of Gamma_0(N) has a complex structure of
oriented Riemann surface. Possibly make some general remarks about
the different types of complex uniformizations of Riemann surfaces in
general.
* Explain why if ord( L(f,s) ) = 1 ====> Sha_an is in QQ. Doing
all the details is hundreds of page.
William Stein
> Hi,
> Project wise I would like to work on:
> * Write a paper explaining how X_0(N) has a complex structure.
> I.e., explain carefully how the orbits for the extended upper half
> plane under the action of Gamma_0(N) has a complex structure of
> oriented Riemann surface. Possibly make some general remarks about
> the different types of complex uniformizations of Riemann surfaces in
> general.
> that is the case.
I don't think anybody has chosen that project. Sounds good!
William