The generalized Chebotarev problem in higher genus

TL;DR

This paper extends the classical Chebotarev problem to higher genus Riemann surfaces, developing a capacity theory and exploring connections to Padé approximants and Jenkins-Strebel differentials.

Contribution

It introduces a generalized Chebotarev problem on algebraic curves, defining a new capacity concept and analyzing the impact of topology on solutions.

Findings

Defined a new notion of capacity for higher genus surfaces

Linked the problem to Jenkins-Strebel quadratic differentials

Highlighted the role of homotopy classes in solutions

Abstract

We consider the extension to higher genus Riemann surfaces of the classical Chebotarev problem, with a view towards the development of the theory of Pad\'e\ approximants on algebraic curves. To this end we define an appropriate notion of capacity that mimics the standard one, following works of Chirka and of the author and collaborators. The nontrivial topology of the Riemann surface requires further specification of the ``homotopy class'' of the continua in the solution of the Chebotarev problem. We also discuss the relationship of this problem to the theory of Jenkins-Strebel quadratic differentials.