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. I figure that since I'm writing code using python and sage, I might as well do it right and incorporate it into sage. 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) and 2) functionality for quaternion algebras over the field Q(i) rather than Q.
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). The resulting algebra produces a lattice which will be cocompact/uniform and I've implemented the following algorithms: 1) compute the elements of reduced norm one within a ball. 2) compute left multiplication by an element 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). 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 --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---
