Nonlinear steepest descent approach to orthogonality on elliptic curves

TL;DR

This paper develops a nonlinear steepest descent method for analyzing orthogonal functions on elliptic curves, revealing their asymptotic behavior through Riemann--Hilbert problems and advanced geometric tools.

Contribution

It introduces a novel approach combining nonlinear steepest descent and Riemann surface theory to study orthogonality on elliptic curves, advancing the analysis of Padé-like approximations.

Findings

Asymptotic behavior of orthogonal functions characterized for large degree

Application of Riemann--Hilbert problem framework to elliptic curve orthogonality

Identification of challenges in steepest descent on Riemann surfaces

Abstract

We consider the recently introduced notion of denominators of Pad\'e--like approximation problems on a Riemann surface. These denominators are related as in the classical case to the notion of orthogonality over a contour. We investigate a specific setup where the Riemann surface is a real elliptic curve and the measure of orthogonality is supported on one of the two real ovals. Using a characterization in terms of a Riemann--Hilbert problem, we determine the strong asymptotic behaviour of the corresponding orthogonal functions for large degree. The theory of vector bundles and the non-abelian Cauchy kernel play a prominent role even in this simplified setting, indicating the new challenges that the steepest descent method on a Riemann surface has to overcome.