Håkan Granath: Heun functions and quaternionic modular forms

The Shimura curve of discriminant 10 is uniformized by a subgroup of an arithmetic (2,2,2,3) quadrilateral group. Hence its uniformization is related to a class of special functions called Heun functions. In the talk I will give an introduction to these concepts, present the differential structure of the ring of modular forms for the Shimura curve, and show how one can relate the ring generators to explicit Heun functions for the quadrilateral group.  Furthermore I will describe how the Picard–Fuchs equation of the associated family of abelian surfaces has solutions that are modular forms.

As an application of these explicit identifications, I will describe how the exceptional sets of the associated Heun functions can be determined, and say a few words on how exceptional values can be computed.

The talk is based on joint work with Srinath Baba.