final report

[William Stein]

Hello Sage Days 18 participants,

I've written a final report on Sage Days 18. It's about 2 pages long.
Please see attached, and let me know if there is anything you would
like me to change, add, etc.

-- William

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

[Kenneth A. Ribet]

Jol Bellache --> Jo\"el Bella\"\i che

Sutherland learned Cython
Python?

they also compuationally investigated
computationally

Ken

[victor miller]

Hi Ken, Cython is one of the great things that William and company accomplished.  They improved on an earlier things called Pyrex, which was Python plus some declarations (that's a slight over simplification), which could be translated into C, and often lead to speed ups by a factor of over 100.  It also allowed one to interface C libraries to Sage fairly easily (I've done it myself).

Victor

[William Stein]

On Tue, Dec 22, 2009 at 5:12 PM, Kenneth A. Ribet
<ri...@math.berkeley.edu> wrote:
>  Jol Bellache --> Jo\"el Bella\"\i che

Thanks.

> Sutherland learned Cython
> Python?

I think he learned Cython (=compiled Python).

> they also compuationally investigated
> computationally

Thanks.

[Noam Elkies]

William Stein <wst...@gmail.com> writes:

> Hello Sage Days 18 participants,

> I've written a final report on Sage Days 18. It's about 2 pages long.
> Please see attached, and let me know if there is anything you would
> like me to change, add, etc.

Thanks. I see that Ken already beat me to the misspelling of J.B.'s
name [I guess the \"{e} and \"{\i} characters got lost in transit
somewhere, which is liable to cause bellyaching (-;], and Victor and
William beat me to confirming the spelling of Cython. I also note that
function field ABC is not entirely trivial -- for one thing it's a
named theorem (Mason) -- even if it's almost surely much harder than
the original conjecture over Q, let alone arbitrary number fields.

What's this "anabelian invariant" of elliptic curves that Mazur
connected with superelliptic point counting? Looks interesting
(especially if it's actually computable, since "anabelian" anything
usually looks far removed from anything one can actually compute),
and I don't think I've seen it before.

Before and during the conference we exchanged some e-mails on
computing Heegner points on factors of J_0(N) of dimension >1,
and I sent you some data I computed c.1990 about the special case of
X_0(p)/w_p when that curve has genus 2 or 3 (which accounts for
the first dozen or so examples of a factor of J_0(N) of dimension >1
with non-torsion Heegner points). Was anything more done about
this problem?

NDE

P.S. I see that it's already more like 2.5 pages long...

[Kiran Kedlaya]

On Tue, Dec 22, 2009 at 8:51 PM, Noam Elkies <elk...@math.harvard.edu> wrote:

What's this "anabelian invariant" of elliptic curves that Mazur
connected with superelliptic point counting?  Looks interesting
(especially if it's actually computable, since "anabelian" anything
usually looks far removed from anything one can actually compute),
and I don't think I've seen it before.

You start with an elliptic curve E, view it as an etale cover of itself via the multiplication-by-2 isogeny, then make a quadratic cover branched over the 2-torsion points. In equations, if E has the Weierstrass form y^2 = P(x), you've just made y^4 = P(x). The point is that there is no reason for this construction to respect isogenies, and in fact already in some small examples it distinguishes some isogenous but nonisomorphic curves.

The "anabelian" modifier is meant to evoke Grothendieck's notion of "anabelian geometry". In this context, it would suggest that if you are looking at a family of hyperbolic curves (i.e., curves of negative Euler characteristic--in this case, look at  E minus the origin), you should be able to distinguish them up to isomorphism by comparing the zeta functions of *all* of their etale covers. This of course is not useful in practice; what would be useful would be a bound on how many covers you need. (A good enough such bound would also resolve Manindra Agrawal's question about finding a deterministic GRH-free polynomial time algorithm for extracting square roots in F_p.)

Kiran

[Noam Elkies]

I asked:

< What's this "anabelian invariant" of elliptic curves that Mazur
< connected with superelliptic point counting? Looks interesting
< (especially if it's actually computable, since "anabelian" anything
< usually looks far removed from anything one can actually compute),
< and I don't think I've seen it before.

Kiran Kedlaya <ksk...@gmail.com> replies:

> You start with an elliptic curve E, view it as an etale cover
> of itself via the multiplication-by-2 isogeny, then make a

> quadratic cover branched over the 2-torsion points. [...]

> The "anabelian" modifier is meant to evoke Grothendieck's notion of

> "anabelian geometry". [...]

Thanks. Meanwhile William sent the link

http://wiki.sagemath.org/dayscambridge2/sprints#ComputeFrobeniuseigenvaluesforabunchofcurvestoillustrateKatz-Sarnak

where Barry is quoted as saying that "my worry is that one won't really
see anything terribly interesting if one works only with N=2; but maybe
when one works with N=3" (when one takes a cover branched at the
N-torsion points), and Barry writes that this is more properly
"metabelian" than "anabelian".

> [...] what would be useful would be a bound on how many covers

> you need. (A good enough such bound would also resolve
> Manindra Agrawal's question about finding a deterministic
> GRH-free polynomial time algorithm for extracting square roots
> in F_p.)

This question well predates Agrawal's work on primality testing...

NDE

[David Roe]

Dequehen is two words: de Quehen.
David

[Jared Weinstein]

On the second page, line 9:

a the --> at the

On the second page, line 17:

functions fields --> function fields

On the fourth line from the end:

compuationally  --> computationally

[Kiran Kedlaya]

Just for completeness, I ran one of the tutorials also (basic Python and Sage usage).

Kiran

On Tue, Dec 22, 2009 at 7:50 PM, William Stein <wst...@gmail.com> wrote: