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 --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
