Computational Algebra Seminar: Voight -- Congruence subgroups of triangle groups

We study three-point covers of the projective line whose Galois group is        
either PSL_2(FF_q) or PGL_2(FF_q). We construct these covers by isolating      
certain subgroups of hyperbolic triangle groups which we call "congruence"      
subgroups. These groups include the classical congruence subgroups of          
SL_2(ZZ), Hecke triangle groups, and 19 families of Shimura curves              
associated to arithmetic triangle groups. We determine the field of moduli     
of the curves associated to these groups and thereby realize the above          
groups regularly as Galois groups in many cases over explicitly given           
abelian number fields. This is joint work with Pete L. Clark.