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

机译:自守函数和Schottky-Klein素函数的Riemann-Hilbert问题

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摘要

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.
机译:柯西核类似物的构造对于解决紧Riemann曲面上的Riemann–Hilbert问题至关重要。 Cauchy内核的公式可以作为肖特基组元素的无穷和给出,并且该和通常用于显式评估内核。在本文中,根据相关的肖特基双核的肖特基-克莱因素函数,推导出了柯西核的拟自晶类似物的新公式。在肖特基组上的无限和不是绝对收敛的情况下,该公式为寻找评估柯西核类似物的新方法打开了大门。讨论了该结果在对称不等式函数具有不连续系数的Riemann-Hilbert问题解中的应用。

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