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 -~----------~----~----~----~------~----~------~--~---
