Arithmetic Fuchsian Groups and Shimura Curves#

In this chapter, we document algorithms for arithmetic Fuchsian groups. Let \(F\) be a totally real number field. Let \(A\) be a quaternion algebra over \(F\), a central simple algebra of dimension \(4\) (see Chapter Quaternion Algebras), such that \(A\) is split at exactly one real place, corresponding to the injective map \(\iota_\infty:A \to M_2({\mathbb{R}})\). Let \({\cal O}\) be a maximal order in \(A\) (see also Section Orders), let \({\cal O}^*\) denote the group of elements of \({\cal O}\) of reduced norm \(1\), and let \(\Gamma(1)=\iota_\infty({\cal O}^*)/\{\pm 1\} \subset PSL_2({\mathbb{R}})\). An arithmetic Fuchsian group \(\Gamma\) is a discrete subgroup of \(PSL_2({\mathbb{R}})\) which is commensurable with \(\Gamma(1)\) (for some choice of \(F\) and \(A\)).

The classical case of the modular groups corresponds to \(F={\mathbb{Q}}\), \(A=M_2({\mathbb{Q}})\), \({\cal O}=M_2({\mathbb{Z}})\), and \(\Gamma(1)={\operatorname{PSL}}_2({\mathbb{Z}})\). Specialized algorithms for this case apply, and they are treated in detail in Chapter Congruence Subgroups of \({\operatorname{PSL}}_2({\mathbb{R}})\). To exclude this case, we assume throughout that \(A\) is a division ring, or equivalently \(A \not\cong M_2({\mathbb{Q}})\).

The group \(\Gamma\) acts properly and discontinuously on the upper half-plane \({\mathbb{H}}\), and the quotient \(\Gamma \backslash {\mathbb{H}}= X(\Gamma)\) can be given the structure of a compact Riemann surface, called a Shimura curve.

We exhibit methods for computing with arithmetic Fuchsian groups \(\Gamma\), including the basic invariants of \(\Gamma\) and a fundamental domain for \(X(\Gamma)\). We provide further specialized algorithms for triangle groups. Along the way, we also present an interface for computing with the unit disc, parallel to that for the upper half-plane (see Chapter Congruence Subgroups of \({\operatorname{PSL}}_2({\mathbb{R}})\)).

The algorithm used to compute fundamental domains is described in [Voight, 2009]. We recommend the following additional reading concerning the algorithms in this section. For an introduction to Fuchsian groups, see Katok [Katok, 1992] and Beardon [Beardon, 1977], and for their relationship to quaternion algebras, see also Vignéras [Vignéras, 1980]. For a computational perspective on arithmetic Fuchsian groups and Shimura curves, see Alsina-Bayer [Alsina and Bayer, 2004], Elkies [Elkies, 1998], and Voight [Voight, 2005], and for a discussion of triangle groups, see Voight [Voight, 2006].