Algebra Seminar

For certain parameter choices, the classical Gauss hypergeometric and Appell hypergeometric functions give rise to monodromy groups acting discontinuously on the 1 and 2 dimensional complex balls respectively. In the 1-dimensional case, all but finitely many of these monodromy groups are non-arithmetic. In the 2-dimensional case, the list of groups acting discontinuously on the 2-ball is finite. These groups have been studied by Picard-Terada-Deligne-Mostow and the non-arithmetic examples identified. The beautiful properties of arithmetic groups are well-known. But these examples of hypergeometric non-arithmetic groups are also interesting and we discuss some of their features and applications in number theory.