Algebra/Topology seminar

Speaker: Robert Kucharczyk

Title: Modular embeddings for triangle groups

Abstract: Lattices in SL(2,R) have been studied extensively since the 19th century since every algebraic curve can be constructed as the quotient of the upper half plane by such a lattice. From a geometric point of view the simplest of these lattices are the Fuchsian triangle groups; from an algebraic point of view these are rather the arithmetic lattices. Apart from a finite list of exceptions triangle groups are non-arithmetic, but they admit certain natural embeddings into arithmetic lattices of higher rank. In this talk I will explain how such embeddings help us to understand the number-theoretic properties of curves uniformised by subgroups of triangle groups.