Discrete Riemann surfaces based on quadrilateral cellular decompositions

TL;DR

This paper develops a comprehensive theory of discrete Riemann surfaces using quadrilateral cellular decompositions, introducing new concepts like discrete coverings, Abel-Jacobi maps, and a discrete Riemann-Roch theorem.

Contribution

It introduces novel notions such as branched coverings, the discrete Riemann-Hurwitz formula, and a discrete Abel-Jacobi map, expanding the theoretical framework of discrete Riemann surfaces.

Findings

Formulated a discrete Riemann-Hurwitz formula.

Defined discrete Abel-Jacobi maps and proved related theorems.

Extended the theory to include branched coverings and discrete meromorphic functions.

Abstract

Our aim in this paper is to provide a theory of discrete Riemann surfaces based on quadrilateral cellular decompositions of Riemann surfaces together with their complex structure encoded by complex weights. Previous work, in particular of Mercat, mainly focused on real weights corresponding to quadrilateral cells having orthogonal diagonals. We discuss discrete coverings, discrete exterior calculus, and discrete Abelian integrals. Our presentation includes several new notions and results such as branched coverings of discrete Riemann surfaces, the discrete Riemann-Hurwitz Formula, double poles of discrete one-forms and double values of discrete meromorphic functions that enter the discrete Riemann-Roch Theorem, and a discrete Abel-Jacobi map.