Thanks John!
On Jul 10, 4:04 am, John Cremona <john.crem...@gmail.com> wrote:
> I forwarded the whole thread to sage-nt (which I see you have joined,
> Lenny) and changed the title there,
>
> John
>
> 2009/7/10 Leonard Foret <a314s...@gmail.com>:
>
>
>
> > Is possible to change the name of this discussion? I made a mistake,
> > the lattice is in SL(2, CC) and not SL(2, Z[i]).
>
> > Lenny
>
> > On Jul 9, 8:02 pm, Leonard Foret <a314s...@gmail.com> wrote:
> >> Excellent, will do. That was my original idea, but I was thrown off a
> >> bit by the request for membership. Anyway, the request went through
> >> so I'll re-post this there.
>
> >> Thanks,
> >> Lenny
>
> >> On Jul 9, 4:24 am, John Cremona <john.crem...@gmail.com> wrote:
>
> >> > It's clear what your algebra is: over the base field K=Q(i) it's the
> >> > quaternion algebra with parameters 2,5.
>
> >> > I think that sage-nt would be a better forum for this than sage-devel.
> >> > Ask to join (athttp://groups.google.co.uk/group/sage-nt).
>
> >> > John
>
> >> > 2009/7/9 Leonard Foret <a314s...@gmail.com>:
>
> >> > > The problem is about finding co-compact lattices in SL(2, C) by using
> >> > > quaternion algebras.
>
> >> > > The example we are working out now is based on the Quaternion algebra
> >> > > over Q(i) defined by the quadratic extension Q(i)[X]/(X^2 - 2) and
> >> > > additional (non-commutative) relation s^2 = 5.
>
> >> > > We need the algebra to be over Q(i) - that's important for the
> >> > > construction of the lattice.
>
> >> > > The lattice is constructed by collecting the elements of the algebra
> >> > > of reduced norm 1 with and with coefficients in the ring of Gaussian
> >> > > integers. The reduced norm is defined by using the left multiplication
> >> > > by an element on the algebra.
>
> >> > > It is possible to figure out how long I should compute before getting
> >> > > all the generators of the lattice, but the idea is to avoid the use of
> >> > > the corresponding theory for this example, and try to work out an
> >> > > approach for any particular case.
>
> >> > > Finding the elements of norm one is related to solving Diophantine
> >> > > equations over Z. Do you know if there is a software for finding
> >> > > generators of groups (like the one we are dealing with in our
> >> > > example)?
>
> >> > > On Jul 7, 9:41 pm, William Stein <wst...@gmail.com> wrote:
> >> > >> On Sat, Jul 4, 2009 at 8:39 PM, Leonard Foret<a314s...@gmail.com>
> >> > >> wrote:
>
> >> > >> > Hello all,
>
> >> > >> > This is my first time in sage-devel. I have a project with a
> >> > >> > professor til the end of August to construct cocompact/uniform
> >> > >> > lattices on SL2(Z[i]) basically by quaternion algebras.
>
> >> > >> What is a "lattice on SL2(Z[i])"?
>
> >> > >> > I figure that
> >> > >> > since I'm writing code using python and sage, I might as well do it
> >> > >> > right and incorporate it into sage.
>
> >> > >> Yes, definitely.
>
> >> > >> > The two contributions I could
> >> > >> > make to sage would be 1) to redo a polished version of the code
> >> > >> > which
> >> > >> > computes the generators of elements of reduced norm one within a
> >> > >> > certain radius for an explicit example (hopefully extend it to a
> >> > >> > general skew field/quaternion algebra)
>
> >> > >> Good.
>
> >> > >> > and 2) functionality for
> >> > >> > quaternion algebras over the field Q(i) rather than Q.
>
> >> > >> What functionality do you want to add?
>
> >> > >> > What I have right now are some python/sage code which looks at the
> >> > >> > quaternion algebra over Q(i) given by the field extension Q(i)(sqrt
> >> > >> > (2)) over Q(i) and the added relation j^2 = 5, (similar to the
> >> > >> > construction that's already implemented over Q).
>
> >> > >> As a non-commutative ring, isn't that precisely exactly the same
> >> > >> thing as
> >> > >> the quaternion algebra
>
> >> > >> sage: R.<i,j,k> = QuaternionAlgebra(-1,5)
> >> > >> sage: R
> >> > >> Quaternion Algebra (-1, 5) with base ring Rational Field
>
> >> > >> already in Sage? Is the point just that you're viewing it differently
> >> > >> as a quadratic
> >> > >> extension of Q(i)?
>
> >> > >> > The resulting algebra
> >> > >> > produces a lattice which will be cocompact/uniform and I've
> >> > >> > implemented the following algorithms:
>
> >> > >> Which lattice? In what space?
>
> >> > >> > 1) compute the elements of reduced norm one within a ball.
>
> >> > >> elements in what?
>
> >> > >> > 2) compute left multiplication by an element
>
> >> > >> left multiplication on what?
>
> >> > >> > 3) compute a norm for these elements (that is, by a norm for the
> >> > >> > matrix computed in 2)
>
> >> > >> > When the elements of reduced norm one are considered with
> >> > >> > multiplication, they form a group and the following algorithm is
> >> > >> > applicable:
> >> > >> > 4) compute generators for the elements of reduced norm one.
>
> >> > >> > The trouble with algorithm 4 and 1, is that it's by complete brute
> >> > >> > force to the point where the algorithm works but I don't know how
> >> > >> > long
> >> > >> > it would take to find all of them (for 1 there's infinitely many).
>
> >> > >> Since you seem to be doing this for exactly the 1 single ring
> >> > >> Q(i)(sqrt(2)), shouldn't you know?
>
> >> > >> > As for adding functionality to Quaternion Algebra I would like to
> >> > >> > work
> >> > >> > on the following:
> >> > >> > 1) extend .is_division_algebra() to the base field Q(i).
> >> > >> > 2) .is_anisotropic()
> >> > >> > 3) and any others.
>
> >> > >> > I'm a beginning graduate student at the Florida International
> >> > >> > University and am working closely with a professor there. If anyone
> >> > >> > is interested or can offer any advice (books, articles to read,
> >> > >> > ideas
> >> > >> > for the algorithms, etc), it would be well received and I'll
> >> > >> > implement
> >> > >> > them immediately.
>
> >> > >> > Thanks in advance!
> >> > >> > Leonard Foret
>
> >> > >> --
> >> > >> William Stein
> >> > >> Associate Professor of Mathematics
> >> > >> University of Washingtonhttp://wstein.org
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