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I was wondering if anyone had some ideas (books, papers, experience) on how to explicitly compute generators for the elements of a quaternion algebra, $Q$, with reduced norm $1$. I'm trying to implement some software using a naive approach, but the results have been less than fruitful. As an example

Let $K = {\mathbb Q}(I)$ (where $I^2=-1$) and let $$Q = \{ z_1 + z_2 i + z_3j + z_4k\;|\;z_i\in K,\;i^2 = 2,\;j^2=5,\;ij= k\}$$ Also let $$ D = \{ z_1 + z_2 i + z_3j + z_4k\in Q\;|\; q\text{ has reduced norm 1},\; z_i\in \mathbb Z[I]\} $$ According to general theory, $Q$ is a skew field and the elements $D$ form a cocompact lattice in $\text{SL}(2,{\mathbb C})$. So by a paper by A. Borel and Harish-Chandra, one can find a finite number of generators for this group $D$ (and in general, any cocompact lattice).

Any advice on finding generators for cocompact lattices in general would be hugely helpful. Thanks in advance!

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  • $\begingroup$ See my answer mathoverflow.net/questions/132723 . This gives a very general recipe. Maybe one can do better with the group you have. $\endgroup$
    – Misha
    Oct 15, 2013 at 13:11
  • $\begingroup$ Thanks for the comment Misha, this is great information! $\endgroup$
    – Lenny
    Oct 15, 2013 at 18:35
  • $\begingroup$ I think you mean $Q$ is a skew-field and $D$ is a cocompact lattice. $\endgroup$
    – Aurel
    Oct 16, 2013 at 12:49
  • $\begingroup$ Oops sorry for the type-o. Thank Aurel $\endgroup$
    – Lenny
    Oct 19, 2013 at 15:06

1 Answer 1

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This is precisely what my Magma package does, if the base field $K$ has at most one complex place. The algorithm is described in this paper.

In the example you give, the set $\{ -7 + (-4I + 5)i + (24I + 6)j + (17I + 4)k, -2I + 3Ij + 2Ik, 3 + Ii + 4Ij + 3Ik, 2 - 4Ii + 11Ij + 8Ik, -I - Ii, -I + 2i - 4j - 3k, 2 + Ii - 7Ij - 5Ik, -3 - Ii - 10j - 7k, -6 + 13j + 9k, -3I - 4j - 3k, (-4I + 1) + (2I + 4)i + (6I - 14)j + (4I - 10)k, 4I + Ii + (10I + 7)j + (7I + 5)k, I + 2i + 10Ij + 7Ik, -4I + 3i + 13Ij + 9Ik, -I - 3i + 14j + 10k, (4I - 1) + (-2I - 4)i + (6I - 14)j + (4I - 10)k, (-4I - 1) + (-2I + 4)i + (-6I - 14)j + (-4I - 10)k \}$ generates the group $D$.

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  • $\begingroup$ Hi Aurel, do you know if the package you made will work with the student edition of Magma (Linux version)? $\endgroup$
    – Lenny
    Oct 19, 2013 at 20:45
  • $\begingroup$ From what they write about the student version it should work, but the memory restriction may prevent you from doing very large computations. However, I have not tried running my code on the student version. If you do, please send me some feedback ! $\endgroup$
    – Aurel
    Oct 20, 2013 at 9:17
  • $\begingroup$ Hi Aurel, I successfully ran your package with the Student version of Magma! However, I'm having trouble figuring out the commands to give the generators for the example above. $\endgroup$
    – Lenny
    Nov 3, 2013 at 1:09
  • $\begingroup$ Did you read the README file ? ;-) _<x> := PolynomialRing(Rationals()); K<I> := NumberField(x^2+1); Q<i,j,k> := QuaternionAlgebra<K|2,5>; O := Order([1,i,j,k]); Genes := NormalizedBasis(O); This is the most basic use (not the fastest and gives more generators than needed). Email me if you want some more detailed help. $\endgroup$
    – Aurel
    Nov 3, 2013 at 23:21
  • $\begingroup$ I sent an email, did you receive it? Running the code as in the example.m file, I get 32 generators (double the amount you have posted). I ran the lines you have above, but with Order = MaximalOrder(Q); and Genes := NormalizedBasis(O : Maple:= true); How can I reduce the number of generators further? $\endgroup$
    – Lenny
    Nov 12, 2013 at 19:28

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