Riemann–Hilbert Problem for Automorphic Functions and the Schottky–Klein Prime Function

Abstract

The construction of analogues of the Cauchy kernel is crucial for the solution of Riemann–Hilbert problems on compact Riemann surfaces. A formula for the Cauchy kernel can be given as an infinite sum over the elements of a Schottky group, and this sum is often used for the explicit evaluation of the kernel. In this paper a new formula for a quasi-automorphic analogue of the Cauchy kernel in terms of the Schottky–Klein prime function of the associated Schottky double is derived. This formula opens the door to finding new ways to evaluate the analogue of the Cauchy kernel in cases where the infinite sum over a Schottky group is not absolutely convergent. Application of this result to the solution of the Riemann–Hilbert problem with a discontinuous coefficient for symmetric automorphic functions is discussed.