School of Mathematics and Statistics
Number Theory Seminar 

Over the complex numbers, a Shimura curve is a Riemann surface which is uniformized by an arithmetic Fuchsian group. Such curves in fact have a much richer structure: they are (coarse) moduli spaces for certain abelian varieties with extra endomorphisms. In this discussion, we present an account of this theory which will focus on two cases of particular interest: curves over the rational numbers and curves uniformized by arithmetic triangle groups. 