Ruminations on Hejhal's theorem about the Bergman and Szego kernels

Abstract

We give a new proof of Dennis Hejhal's theorem on the nondegeneracy of the matrix that appears in the identity relating the Bergman and Szego kernels of a smoothly bounded finitely connected domain in the plane. Mergelyan's theorem is at the heart of the argument. We explore connections of Hejhal's theorem to properties of the zeroes of the Szego kernel and propose some ideas to better understand Hejhal's original theorem.