This is the opposite to my last question case.

Let $F$ be a totally real number field, $R$ is a quaternion algebra over $F$ unramified in at least one infinite place of $F$. Let $\mathcal{O}⊂R$ be an order.

Can I explicitly compute the rank and generators of the group of units $\mathcal{O}^{\times}$ in terms of, say, some basis in $R/F$?

It seems that the rank is related to the corresponding Shimura curve, but I do not see how this helps.

  • $\begingroup$ What do you mean by rank of this noncommutative group? $\endgroup$ – Aurel Sep 26 '16 at 22:00
  • $\begingroup$ @Aurel Just the minimal number of generators $\endgroup$ – SashaP Sep 26 '16 at 23:03
  • 1
    $\begingroup$ Ok, then in the Fuchsian case (only one split real place) maybe you can compute the rank without actually computing generators, using standard presentations of Fuchsian groups. I think this is also discussed in Voight's article I mention below. $\endgroup$ – Aurel Sep 26 '16 at 23:55

If there is only one split real place, you can use the algorithm described by John Voight in Computing fundamental domains for Fuchsian groups. This algorithm is available in Magma.


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